From 3aa483b31c12d6a3afb1215ce6188aca8c314e72 Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
Date: Wed, 30 Dec 2020 11:03:54 +0000
Subject: Update on Overleaf.

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+
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+
+\begin{document}
+
+\title{{\bf COVER LETTER \\`Complex complex landscapes'}}
+%\footnote{{\bf Key-words}: }
+%}}
+
+\author{
+Jaron Kent-Dobias
+ and 
+Jorge Kurchan
+}
+
+\maketitle
+
+
+
+\vspace{1.cm}
+
+
+The subject of `Complex Landscapes', which started in the spin-glass literature, is concerned with functions (landscapes) of many variables, having a multiplicity of minimums, which are the objects of interest. Apart from its obvious interest for glassy systems, it has found a myriad applications  in  many domains:  Computer Science, Ecology, Economics, Biology \cite{mezard2009information}.
+
+In the last few years, a renewed interest has developed for landscapes for which the variables are complex. There are a few reasons for this: {\em i)} in Computational Physics, there is the main obstacle of the `sign problem', and a strategy has emerged to attack it deforming the sampling space into complex variables. This is a most natural and promising path, and any progress made will have game-changing impact in solid state physics and lattice-QCD \cite{Cristoforetti_2012_New,Scorzato_2016_The}.
+{\em ii)} At a more basic level, following the seminal work of E. Witten \cite{Witten_2010_A,Witten_2011_Analytic}, there has been a flurry of activity concerning the very definition of quantum mechanics, which requires also that one move into the complex plane. 
+
+In all these cases, just like in the real case, one needs to know the structure of the `landscape', where are the saddle points and how they are connected, typical questions of `complexity'.
+However, to the best of our knowledge, there are no studies extending the methods of the theory of
+complexity to
+complex variables. 
+We believe our paper will open a field that may find 
+numerous applications and will widen our theoretical view of complexity in general.
+
+
+\bibliography{bezout}
+
+
+\end{document}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\section{The Kipnis-Marchioro-Presutti model}
+
+Consider the following process: 
+\begin{itemize}
+\item
+choose a pair of neighbouring sites and completely
+exchange energy between them
+\item
+if the site is one of the borders, exchange completely energy with the bath.
+\end{itemize}
+each choice with probability $1/(N+1)$. From here onwards, we shall denote
+$\tau$ a large time, sufficient for any two-site thermalisation.
+
+The evolution operator in one step is:
+\begin{eqnarray}
+U &=& \frac{1}{N+1} \left[ e^{-\tau L_1^*} +  e^{-\tau L_N^*} + \sum_{i=1}^{N-1}   e^{-\tau L^*_{i,i+1}} \right]
+\nonumber \\
+&=& \frac{1}{N+1} \left[ e^{-2\tau (T_1 K^-_1 + K^o_1 + k) } +  e^{-2\tau(T_L K^-_L + K^o_L +k)  }
+ + \sum_{i=1}^{N-1}   e^{ \frac{-\tau}{k}
+(K^+_i K^-_{i+1} + K^-_i K^+_{i+1} - 2 K^o_i K^o_{i+1}
++ 2k^2 )} \right] \nonumber \\
+~
+\end{eqnarray}
+and the dynamics after $n$ steps is given by $U^n$.
+Because we are considering large $\tau$, the terms in the sums are  in fact projectors
+onto the lowest eigenvalues of the exponents. We shall however keep the notation as it is
+in order to stress the symmetry of the bulk terms.
+
+Let us now show that - at the level of energies - this dynamics yields the KMP process 
+{\em for $k=\frac{1}{2}$, that is $m=2$}.
+Consider first a general $m$, and two neighbouring sites of coordinates $x = \{x_\alpha\}_{\alpha=1,\ldots,m}$, 
+$y=\{y_\alpha\}_{\alpha=1,\ldots,m}$.
+If they are completely thermalised, it means that (cfr (\ref{bb}):
+the joint probability density satisfies
+\begin{equation}
+\left(x_{\alpha}
+\frac{\partial}{\partial y_{\beta}} -
+y_{\beta}\frac{\partial}{\partial x_{\alpha}}
+  \right) p(x,y)=0
+\end{equation}
+It is easy to see that this may happen if and only if
+\begin{equation}
+p(x,y)= p[ \sum_\alpha (x_\alpha^2+y_\alpha^2)]
+\end{equation}
+In particular let us consider the microcanonical measure
+\begin{equation}
+p(x,y)= \delta[ \sum_\alpha (x_\alpha^2+y_\alpha^2)-\epsilon ]
+\end{equation}
+Defining new random variables $\epsilon_1$ and $\epsilon_2$ 
+as the energies of the neighboring sites
+\be
+\epsilon_1 = \sum_\alpha x_\alpha^2
+\ee
+\be
+\epsilon_2 = \sum_\alpha y_\alpha^2
+\ee
+then their joint probability density will be
+\begin{equation}
+p(\epsilon_1,\epsilon_2) = \frac{S_m^2}{4} \delta(\epsilon_1+\epsilon_2-\epsilon)
+\epsilon_1^{\frac{1}{2}-1} \epsilon_2^{\frac{1}{2}-1}
+\end{equation}
+where $S_m$ denotes the surface of the unit sphere in $m$ dimension
+\be
+S_m = \frac{m \pi^{m/2}}{\Gamma(\frac{1}{2}+1)}
+\ee
+{\em This yields a flat distribution for $m=2$, i.e. the KMP model.}
+
+
+
+
+\section{Dual model}
+
+
+The expectation value of an observable at time $t$, starting from an initial
+distribution $|init\rangle$ is:
+
+
+\begin{equation}
+<O> = \langle - | O e^{-Ht} | init \rangle
+\end{equation}
+where $\langle - |$ is a constant.
+Taking the adjoint $ x_i \to x_i$, $\partial_i \to -\partial_i$:
+\begin{equation}
+<O> = \langle - | O e^{-Ht} | init \rangle= \langle init|  e^{-H^\dag t} O |- \rangle
+\end{equation}
+where $H^\dag(K^\pm, K^o)=H( K^\pm, -K^o)$ (because of the change of signs of the derivatives)
+\begin{eqnarray}
+-H^\dag&=& \frac{4}{1} \sum_i \left(
+K^+_i K^-_{i+1} + K^-_i K^+_{i+1} - 2 K^o_i K^o_{i+1}
++ \frac{m^2}{8} \right)
+\nonumber\\
+&+&2 \left(T_1 K^-_1 + K^o_1 + \frac{1}{4}\right)
++2 \left(T_L K^-_L + K^o_L +\frac{1}{4}\right)
+\end{eqnarray}
+In particular, for the generating function we had chosen
+ \begin{equation}
+ O |- \rangle = \Pi_i \frac{x_i^{2 \xi_i}}{(2\xi_i -1)!!}|-\rangle=|\xi_1,...,\xi_N\rangle
+\end{equation}
+
+Considered as an operator acting on `particle number', as counted by $K^o$, $H^\dag$ does not
+conserve the probability.
+The trick we used can be expressed as follows: introduce the particle number $\xi_o$ and $\xi_{N+1}$ 
+and the operators $A^+_o$ and $A^+_{N+1}$, which create particles in boundary sites with unit rate.
+We consider now the {\em enlarged} process generated by
+\begin{eqnarray}
+-H^{dual}&=& \frac{4}{1} \sum_i \left(
+K^+_i K^-_{i+1} + K^-_i K^+_{i+1} - 2 K^o_i K^o_{i+1}
++ \frac{m^2}{8} \right)
+\nonumber\\
+&+&2 \left(A^+_o K^-_1 + K^o_1 - \frac{1}{4}\right)
++2 \left( A^+_{N+1} K^-_N + K^o_N -\frac{1}{4}\right)
+\end{eqnarray}
+which conserves ({\it seems}) particle number and probability.
+We wish to prove that:
+
+\begin{eqnarray}
+<O> &=& \langle init| e^{-H^\dag t} |\xi_1,...,\xi_N \rangle \nonumber \\
+&=& \sum_{\xi_o,\xi_{N+1}}
+  T_1^{\xi_o} T_{L}^{\xi_{N+1}} \langle \xi_o \xi_{N+1} | \otimes \langle
+  init| e^{-H^{dual} t} |\xi_1,...,\xi_N \rangle \otimes
+  |\xi_o=0,\xi_{N+1}=0 \rangle \nonumber \\
+\label{ggg}
+\end{eqnarray}
+
+
+I think the proof is obvious, because developing the exponential of $H^{dual}$ all the $A^+$ can be
+collected because they commute with everything else, and the experctation value
+\begin{equation}
+\sum_{\xi_o}  T_1^{\xi_o}  \langle \xi_o  |[A^+_o]^r  |\xi_o=0 \rangle = T_1^r
+ \end{equation}
+just puts back as many $T$'s as necessary.
+
+I do not know exactly how to use (\ref{ggg}) in general, but in the large time limit the evolution
+voids the chain of particles
+
+
+\section{Dual of KMP}
+
+I think that the argument runs through without changes if we use $U$ defined for the KMP model.
+We just have to note that each term corresponds to an evolution of two sites (or a site and the bath)
+and so in the dual it corresponds to sharing the particles between those two sites, or emptying
+the sites at the borders.
+
+{\bf: NOTE by Cristian}
+
+We can check that the duality function chosen in the original paper by KMP
+do coincide with the duality function of our process for $m=2$ (and the random
+variables are the energies).
+Indeed we start from
+\be
+f(x,\xi) = \prod_i (\sum_{\alpha} x_{i,\alpha}^2)^{\xi}
+\ee
+When the bath have equal temperature (let's us choose T=1) then the stationary 
+measure is 
+\be
+\pi(x) = \prod_i \frac{1}{(2\pi)^{m/2}} \exp\left(-\sum_{\alpha}\frac{x_{i,\alpha}^2}{2}\right)
+\ee
+Let us focus on a fixed $i$ (that is in this short computation we write $x$ for $x_i$). 
+We have
+\begin{eqnarray}
+\E(f(x,\xi)) 
+&=&
+\int dx_1 \cdots \int dx_m (x_1^2+\ldots + x_m^2)^{\xi} \exp-\left(\frac{x_{1}^2}{2}+\ldots+\frac{x_{1}^2}{2}\right) 
+\nonumber \\
+& = & 
+\int dr S_m r^{2\xi} \exp-\left(\frac{r^2}{2}\right)
+\nonumber \\
+& = &
+\frac{\frac{1}{2}\Gamma(\frac{1}{2}+\xi)}{\Gamma(\frac{m}{2}+1)} 2^\xi
+\nonumber \\
+\end{eqnarray}
+Special cases:
+\begin{itemize}
+\item $m=1$
+
+$$
+\E(f(x,\xi)) = (2\xi-1)!!
+$$
+where one uses that $\Gamma(\frac{1}{2}+\xi)= \frac{\sqrt{\pi}(2\xi-1)!!}{2^{\xi}}$ and $\Gamma(\frac{3}{2}) = \frac{\sqrt{\pi}}{2}$
+\item $m=2$
+
+$$
+\E(f(x,\xi)) = \xi! 2^\xi
+$$
+where one uses that $\Gamma(1+\xi)= \xi!$ and $\Gamma(2) = 1$.
+Thus, if one defines the energies as 
+$$
+\epsilon_i = \sum_{\alpha}\frac{x_{i,\alpha}^2}{2}
+$$
+one recover the choice of KMP for the dual function
+$$
+O(\epsilon_i,\xi) = \prod_i \frac{\epsilon_i^{\xi_i}}{\xi_i!}
+$$
+\end{itemize}
+
+
+
+
+
+
+\section{ Dual of SEP: here goes an outline of how to proceed for the SSEP}
+
+
+\be
+H=-L_{SEP}^*
+\ee
+\begin{eqnarray}
+L^*_{SEP} &=& \frac{1}{j}
+  \sum_i  \left(J^+_i J^-_{i+1} + J^-_i J^+_{i+1} + 2 J^o_i J^o_{i+1}
+ - 2 j^2  \right)\\
+&+&\alpha (J^-_1 - J^o_1-j) + \gamma (J^+_1 + J^o_1-j)
++ \delta (J^-_L - J^o_L-j) + \beta (J^+_L + J^o_L-j)\nonumber
+\end{eqnarray}
+The factor $1/j$ is analogous to the factor $1/m$ in (\ref{bb}).
+The operators $J^+_i, J^-_i, J^o_i$ act on the Hilbert space
+ corresponding to  $0 \le r \le n$  particles per site $\otimes_i |r\rangle_i$
+as follows:
+\begin{eqnarray}
+J^+_i |r\rangle_i &=& (2j-r)  |r+1\rangle_i \nonumber \\
+ J^-_i |r\rangle_i &=& r  |r-1\rangle_i \nonumber \\
+J^o_i |r\rangle_i &=& (r-j)  |r\rangle_i
+\end{eqnarray}
+
+The conjugation properies are as follows. There is an operator $Q$,
+{\em diagonal in this basis  } (I give the expression below), such that:
+\begin{equation}
+[J^+_i]^\dag = Q[J^-_i]Q^{-1} \qquad [J^-_i]^\dag = Q[J^+_i]Q^{-1}
+\end{equation}
+while $[J^z_i]^\dag=J^z_i= Q[J^z_i]Q^{-1}$.
+
+
+
+The expectation value of an observable at time $t$, starting from an initial
+distribution $|init\rangle$ is:
+
+
+\begin{equation}
+<O> = \langle - | O e^{-Ht} | init \rangle
+\end{equation}
+where $\langle - |$ is a constant.
+As before:
+\begin{eqnarray}
+<O> &=& \langle - | O e^{-Ht} | init \rangle=
+\langle init|  e^{-H^\dag t} O |- \rangle= \nonumber \\
+& & \langle init|Q  e^{-{\bar H} t}  Q^{-1}O   |- \rangle=
+\langle init|Q \;  e^{-{\bar H} t}  Q^{-1}O Q  Q^{-1} |- \rangle
+\end{eqnarray}
+
+
+{\em  $ {\bar H}$ is the same operator as $H$ but with
+$J^+$ substituted by  $J^-$, and vice-versa.}
+Our job is now to make the rotation that will eliminate the $J^+$'s in
+the border terms of $ {\bar H}$.
+
+
+
+
+The transformation is of the form
+\begin{eqnarray}
+e^{\mu J^+} J^+ e^{-\mu J^+}&=&J^+ \nonumber \\
+e^{\mu J^+} J^o e^{-\mu J^+} &=&J^o - \mu J^+ \nonumber \\
+e^{\mu J^+} J^- e^{-\mu J^+} &=& J^- + 2 \mu J^o - \mu^2 J^+
+\end{eqnarray}
+for suitable $\mu$.
+Putting $\mu=-1$, we get that {\bf the bulk term is left invariant,
+precisely because of the SU(2) symmetry}. The boundary terms {\bf of $\bar H$}
+transform further into:
+\begin{eqnarray}
+& e^{\mu J^+_1} \left[ \alpha (J^+_1 - J^o_1-j) + \gamma (J^-_1 + J^o_1-j)
+\right] e^{-\mu J^+_1}= \nonumber \\ & \gamma(J^-_1 + 2 \mu J^o_1 - \mu^2
+J^+_1 +J^o_1 - \mu J^+_1 -j) + \alpha (J^+_1 - J^o_1 + \mu J^+_1 -j)
+= \nonumber \\
+& \alpha(- J^o_1 -j) + \gamma (J^-_1 -J^o_1 -j)
+\label{trans}
+\end{eqnarray}
+which is of the same form we have in the $SU(1,1)$ model.
+The same can be done in the other boundary term.
+
+We thus get:
+\begin{eqnarray}
+<O> &=& \langle - | O e^{-Ht} | init \rangle=
+\langle init|Q \; e^{-{\bar H} t}  Q^{-1}O Q  Q^{-1} |- \rangle \nonumber\\
+&= & \langle init|Q   e^{ \sum_i J^+_i}  e^{-{\bar H_{dual}} t}
+ e^{ -\sum_i J^+_i}  Q^{-1}O Q  Q^{-1} |- \rangle \nonumber \\
+&= & \langle init|Q   e^{ \sum_i J^+_i}  e^{-{\bar H_{dual}} t}
+ e^{ -\sum_i J^+_i}  Q^{-1}O Q e^{ \sum_i J^+_i}
+  e^{ -\sum_i J^+_i}  |- \rangle \nonumber \\
+ &= & \langle init|Q  Q^{-1}  e^{ \sum_i J^+_i}  e^{-{\bar H_{dual}} t}
+ e^{ -\sum_i J^+_i}  Q^{-1}O Q e^{ \sum_i J^+_i}  |-_{dual} \rangle
+\end{eqnarray}
+where we have defined $H_{dual}$  as the transformed Hamiltonian.
+
+We now have to study $ |-_{dual} \rangle \equiv e^{ -\sum_i J^+_i}
+ Q^{-1} |- \rangle$
+Because we know that  terms like those proportional to $\gamma$ and $\alpha$
+anihilate the measure to the left:
+\begin{eqnarray}
+& & \langle - | (J^-_i - J^o_i-j) =0\nonumber \\
+& & \langle - | (J^+_i + J^o_i-j) =0
+\end{eqnarray}
+this implies that in the new variables and following all the transformations
+(cfr (\ref{trans})):
+\begin{eqnarray}
+& & (J^-_i -J^o_i -j)e^{ -\sum_i J^+_i} Q^{-1} |- \rangle= 0 \nonumber \\
+& & ( -J^o_i -j)e^{ -\sum_i J^+_i} Q^{-1} |- \rangle =0
+\end{eqnarray}
+which implies that $( J^o_i +j) |-_{dual} \rangle= J^-_i  |-_{dual} \rangle=0$,
+and this means that
+\begin{equation}
+J^o_i |-_{dual} \rangle =-j |-_{dual}  \rangle
+\end{equation}
+is the vacuum of particles in this base!
+
+All in all we are left with:
+\begin{eqnarray}
+<O> &=&  \langle init|Q \;  e^{ \sum_i J^+_i}
+  e^{-{\bar H_{dual}} t} e^{ -\sum_i J^+_i}  Q^{-1}O Q e^{ \sum_i J^+_i}
+  |-_{dual} \rangle \nonumber \\
+ &=&  \langle init|Q \;  e^{ \sum_i J^+_i}
+  e^{-{\bar H_{dual}} t} {\hat O}
+  |-_{dual} \rangle
+\end{eqnarray}
+where $ {\hat O} \equiv e^{ -\sum_i J^+_i} Q^{-1}O Q e^{ \sum_i
+J^+_i}$.  We have to start with the
+vacuum $ |-_{dual} \rangle$, then apply $ {\hat O} $, (which creates
+particles because it contains many $J^+$'s), and then there is the
+dual evolution. The final configuration has to be overlapped with
+$\langle f| \equiv  \langle init|Q \;  e^{ \sum_i J^+_i}$.
+For large times, there will be no particle left except in the two extra sites
+in the borders.
+
+\section{Constructive approach}
+
+Here I would like to say the following: if I have a modle of transport
+of which I do not know if it has a Dual one, I can proceed as follows.
+I take a small version with no baths and a few sites. I write the
+evolution operator and I diagonalise it numerically.  If there is a
+non abelian group, the eigenvalues will be in degenerate
+multiplets. Hence, if I find multiplets, then very probably there is a
+dual model, if I do not, then there cannot be one.  It would be nice
+to show it with the KMP model with two or three sites.
+
+Another thing is to consider higher groups. $SU(3)$ has already been studied
+for two kinds of particles. We know how to map to a dual in that
+case, if it has not been done yet.
+
+\newpage
+{\bf THIS PART HAS BEEN WRITTEN BY CRISTIAN} 
+
+The aim of this file is to set notation in the two languages.
+Let us focus on duality for the case we already know:
+SU(1,1) model with $k=1/4$. To fix ideas let us consider only
+the bulk part of the system with periodic boundary conditions.
+
+\section{Probabilistic language}
+We have two stochastic Markovian process with continuous time.
+\begin{itemize}
+\item
+\underline{The first process $X(t) \in \R^N$} is given by the Fokker-Planck equation:
+\be
+\frac{dp(x,t)}{dt}  = L^* p(x,t)
+\ee
+where $p(x,t)$ represents the probability density
+for the process $X(t)$, that is
+$$
+p(x,t)dx = Prob (X(t)\in (x,x+dx))
+$$
+and
+\begin{eqnarray}
+L^*p(x,t)
+& = &
+\sum_i L^*_{i,i+1} p(x,t) \noindent\\
+& = &
+\sum_i \left(x_i\frac{\partial}{\partial x_{i+1}} -x_{i+1}\frac{\partial}{\partial x_{i}}\right)^2 p(x,t)
+\end{eqnarray}
+\item
+\underline{The second process $\Xi(t) \in \N^N$} is characterized by the master equation
+\be
+\frac{dP(\xi,t)}{dt} = {\cal L^*} P(\xi,t)
+\ee
+where $P(\xi,t)$ represents the
+probability mass function for the process $\Xi(t)$, that is
+$$
+P(\xi,t) = Prob (\Xi(t) = \xi)
+$$
+and
+\begin{eqnarray}
+{\cal L}^*P(\xi,t)
+& = &
+\sum_i {\cal L}^*_{i,i+1}P(\xi,t) \nonumber \\
+& = &
+\sum_i 2\xi_i \left(1+ 2\xi_{i+1}\right) P(\xi^{i,i+1},t)
++ \left(1+2\xi_i\right)2\xi_{i+1} P(\xi^{i+1,i},t) \nonumber\\
+& & - 2\left(2\xi_i + \frac{1}{2}\right)\left(2\xi_{i+1} + \frac{1}{2}\right) P(\xi,t)
++ \frac{1}{2}P(\xi,t)
+\end{eqnarray}
+and $\xi^{i,j}$ denotes the configuration that is obtained by removing one particle
+at $i$ and adding one particle at $j$.
+\newpage
+\item
+\underline{In general, Duality means the following}:
+there exists functions $O(x,\xi): \R^N \times \N^N \mapsto \R$ such that
+the following equality between expectations for the two processes holds
+\begin{center}
+\fbox{\parbox{9cm}{
+\be
+\E_x( O(X(t),\xi)) =\E_\xi(O(x,\Xi(t)))
+\ee
+}}
+\end{center}
+The subscripts in the expectations denote the initial conditions of the processes:
+$X(0) =x$ on the left and $\Xi(0) = \xi$ on the right.
+More explicitly we have:
+\be
+\int dy O(y,\xi) p(y,t; x,0)  = \sum_{\eta} O(x,\eta) P(\eta,t; \xi,0)
+\ee
+To prove duality it is sufficient to show that
+\be
+\label{main}
+L O(x,\xi) = {\cal L} O(x,\xi)
+\ee
+where $L$, that is working on $x$, is the adjoint of  $L^*$ and ${\cal L}$, that is working on $\xi$,
+is the adjoint of  ${\cal L}^*$.
+Indeed we have:
+\begin{eqnarray}
+\E_x( O(X(t),\xi))
+& = &
+\int dy O(y,\xi) p(y,t; x,0) \\
+& = &
+\sum_{\eta} \int dy O(y,\eta) p(y,t; x,0) \delta_{\eta,\xi} \\
+& = &
+\sum_{\eta} \int dy O(y,\eta) e^{tL^*} \delta(y-x) \delta_{\eta,\xi} \\
+& = &
+\sum_{\eta} \int dy e^{tL} O(y,\eta)  \delta(y-x) \delta_{\eta,\xi} \\
+& = &
+\sum_{\eta} \int dy e^{t{\cal L}} O(y,\eta)  \delta(y-x) \delta_{\eta,\xi}  \\
+& = &
+\sum_{\eta} \int dy O(y,\eta) e^{t{\cal L}^*} \delta(y-x) \delta_{\eta,\xi}  \\
+& = &
+\sum_{\eta} \int dy O(y,\eta) P(\eta,t;\xi,0) \delta(y-x) \\
+& = &
+\sum_{\eta}  O(x,\eta) P(\eta,t;\xi,0)  \\
+& = &
+\E_\xi(O(x,\Xi(t)))
+\end{eqnarray}
+\newpage
+\item
+\underline{For the present case, the proper function to be considered are}
+\be
+\label{Oss}
+O(x,\xi) = \prod_{i} \frac{x_i^{2\xi_i}}{(2\xi_i-1)!!}
+\ee
+Let us check Eq.(\ref{main}) on this choice. We have
+\begin{eqnarray*}
+&&
+L_{i,i+1} O(x,\xi)
+=
+\left(\prod_{k\not\in\{i,i+1\}}  \frac{x_k^{2\xi_k}}{(2\xi_k -1)!!}\right)
+\times
+\\
+&&\left(2\xi_{i+1}(2\xi_{i+1}-1) \frac{x_i^{2\xi_i+2}}{(2\xi_i -1)!!}\frac{x_{i+1}^{2\xi_{i+1}-2}}{(2\xi_{i+1} -1)!!}
+- 2\xi_{i}(2\xi_{i+1}+1) \frac{x_i^{2\xi_i}}{(2\xi_i -1)!!}\frac{x_{i+1}^{2\xi_{i+1}}}{(2\xi_{i+1} -1)!!}
+\right.
+\\
+&&\left.- 2\xi_{i+1}(2\xi_{i}+1) \frac{x_i^{2\xi_i}}{(2\xi_i -1)!!}\frac{x_{i+1}^{2\xi_{i+1}}}{(2\xi_{i+1} -1)!!}
++2\xi_{i}(2\xi_{i}-1) \frac{x_i^{2\xi_i-2}}{(2\xi_i -1)!!}\frac{x_{i+1}^{2\xi_{i+1}+2}}{(2\xi_{i+1} -1)!!}
+\right)
+\\
+\end{eqnarray*}
+which implies
+\begin{eqnarray*}
+L_{i,i+1} O(x,\xi)
+& = &
+\Big(2\xi_{i+1}(2\xi_{i}+1) [O(x,\xi^{i+1,i})-O(x,\xi)]
+\\
+&&
+\;+\;2\xi_{i}(2\xi_{i+1}+1) [O(x,\xi^{i,i+1})-O(x,\xi)]\Big)
+\\
+& = &
+{\cal L}_{i,i+1} O(x,\xi)
+\end{eqnarray*}
+
+\item \underline{How to find the proper normalization?}
+Suppose that we are in the general following situation:
+\begin{itemize}
+\item We have a generator $L$ of a Markov process $X(t)$.
+\item We know its stationary measure $\pi(x)$:
+\be
+L^* \pi(x) = 0
+\ee
+\item We have functions $f(x,\xi)$ for which the following holds:
+\be
+\label{aaa}
+L f(x,\xi) = \sum_{\eta} r(\xi,\eta) f(x,\eta)
+\ee
+with
+\be
+\label{bbb}
+r(\xi,\eta) \ge 0 \qquad  \mbox{if}\quad \xi \neq \eta
+\ee
+\be
+\label{ccc}
+r(\xi,\xi)  \le 0 \qquad  \mbox{if}\quad \xi = \eta
+\ee
+\end{itemize}
+The matrix $r$ resembles the generator of a dual Markov process,
+but it is not because it does not satisfy the condition
+$\sum_{\eta} r(\xi,\eta) = 0$.
+In order to find the generator of the dual process we proceed as
+follows:
+\begin{enumerate}
+\item Define
+\be
+m(\xi) = \int f(x,\xi) \pi(x) dx
+\ee
+\item Define
+\be
+q(\xi,\eta)= m(\xi)^{-1} r(\xi,\eta) m(\eta)
+\ee
+\item Define
+\be
+O(x,\xi) = m(\xi)^{-1} f(x,\xi)
+\ee
+\end{enumerate}
+Then the matrix $q$ can be seen as the generator of the dual Markov process $\Xi(t)$, that is
+\be
+L O(x,\xi) = \sum_{\eta} q(\xi,\eta) O(x,\eta)
+\ee
+with
+\be
+q(\xi,\eta) \ge 0 \qquad  \mbox{if}\quad \xi \neq \eta
+\ee
+\be
+q(\xi,\xi)  \le 0 \qquad  \mbox{if}\quad \xi = \eta
+\ee
+\be
+\sum_{\eta} q(\xi,\eta) = 0
+\ee
+Indeed we have:
+\begin{eqnarray}
+L O(x,\xi)
+&=&
+L m(\xi)^{-1} f(x,\xi) \nonumber \\
+&=&
+m(\xi)^{-1} \sum_{\eta} r(\xi,\eta) f(x,\eta) \nonumber \\
+&=&
+m(\xi)^{-1} \sum_{\eta} m(\xi)q(\xi,\eta) m(\eta)^{-1} m(\eta) O(x,\eta)\nonumber \\
+&=&
+\sum_{\eta} q(\xi,\eta) O(x,\eta)
+\end{eqnarray}
+and
+\begin{eqnarray}
+\sum_{\eta} q(\xi,\eta)
+&=&
+\sum_{\eta} m(\xi)^{-1} r(\xi,\eta) m(\eta)  \nonumber \\
+&=&
+m(\xi)^{-1} \sum_{\eta}  r(\xi,\eta) \int f(x,\eta) \pi(x) dx \nonumber \\
+&=&
+m(\xi)^{-1} \int L f(x,\xi) \pi(x) dx \nonumber \\
+&=&
+m(\xi)^{-1} \int f(x,\xi) L^* \pi(x) dx \nonumber \\
+&=&
+0
+\end{eqnarray}
+
+
+
+\item \underline{Our case}. Among all the invariant measure
+of the $X(t)$ process, namely the normalized function with
+spherical symmetry $p(x) = p(\sum_i x_i^2)$, a special role is
+played by the Gibbs measure
+$$
+\pi(x)
+= \left(\frac{\beta}{2\pi}\right)^{(N/2)} e^{-\beta\sum_i \frac{x_i^2}{2}}
+= \left(\frac{\beta}{2\pi}\right)^{(N/2)} \prod_i e^{-\beta\frac{x_i^2}{2}}
+$$
+which is selected as soon as the system is placed in contact with
+thermal bath working at inverse temperature $\beta$.
+Moreover: If $Z$ is a centered Gaussian, namely $Z\sim N(0,\sigma^2)$,
+then
+$$
+\E(Z^{2n}) = \sigma^{2n} (2n-1)!!
+$$
+If one start from
+$$
+f(x,\xi) = \prod_i x_i^{2\xi}
+$$
+which satisfy (\ref{aaa}),(\ref{bbb}),(\ref{ccc}) and apply
+the previous procedure, one arrives to (\ref{Oss}).
+
+{\bf Remark:} Note that, in applying the procedure, the
+dependence on $\beta$ disappear!!!!
+\end{itemize}
+
+
+\section{Quantum language}
+
+
+Here we start from a quantum spin chain
+$$
+H = - 4 \sum_i \left( K^+_iK^-_{i+1} + K^-_iK^+_{i+1} -2 K^0_iK^0_{i+1} + \frac{1}{8}\right)
+$$
+where the spin $K_i$'s satisfy the SU(1,1) algebra
+\begin{eqnarray}
+\label{commutatorsSU11}
+[K_i^{0},K_i^{\pm}] &=& \pm K_i^{\pm} \nonumber \\
+{[}K_{i}^{-},K_{i}^{+}{]} &=& 2K_i^{0}
+\end{eqnarray}
+We are going to see the Schr\"odinger equation with imaginary time
+\begin{equation}
+\label{schroedinger}
+\frac{d}{dt}|\psi(t) \rangle = -H |\psi(t)\rangle\;.
+\end{equation}
+as the evolution equation for the probability distribution of
+a Markovian stochastic process.
+\begin{itemize}
+\item
+\underline{The Hamiltonian possesses the SU(1,1) invariance}. If we define
+\be
+K^+ = \sum_{i} K_i^+
+\ee
+\be
+K^- = \sum_{i} K_i^-
+\ee
+\be
+K^0 = \sum_{i} K_i^0
+\ee
+we find that
+\be
+[H,K^+] = 0
+\ee
+\be
+[H,K^-] = 0
+\ee
+\be
+[H,K^0] = 0
+\ee
+\item
+\underline{Since $[H,K^+] = 0$} there exist a basis to study the stochastic process associated to
+$H$ where \underline{$K^+$ is diagonal}. We might consider the following representation
+\begin{eqnarray}
+\label{Koper}
+K^+_i &=& \frac{1}{2}  x_{i}^2 \nonumber \\
+K^-_i &=& \frac{1}{2} \frac{\partial^2}{\partial x_{i}^2} \nonumber \\
+K^o_i &=& \frac{1}{4}  \left\{\frac{\partial}{\partial x_{i}} x_{i} +
+ x_{i} \frac{\partial}{\partial x_{i}} \right \}
+\end{eqnarray}
+If we use this representation then
+$$
+H = -L^*
+$$
+and the probability density function for the $X(t)$ process is encoded in
+the state $|\psi(t)\rangle$, namely
+\begin{equation}
+|\psi(t) \rangle = \int dx p(x,t) |x\rangle
+\end{equation}
+where we have introduced the notation $|x\rangle$ to denote a completely
+localized state, that is a vector which together with its transposed
+$\langle x|$ form a complete basis of a Hilbert space and its dual:
+\begin{equation}
+\langle x|x' \rangle = \delta(x-x')
+\end{equation}
+It immediately follows that
+\begin{equation}
+\langle x|\psi(t) \rangle = p(x,t)
+\end{equation}
+To compute expectation with respect to the $X(t)$ process
+we introduce the flat state
+\begin{equation}
+\langle - | = \int dx \;\langle x|
+\end{equation}
+which is such that
+\begin{equation}
+\langle - | x\rangle = 1
+\end{equation}
+Then for any observable $A = A(X(t))$ we have that its expectation value
+at time $t$ can be written as
+\begin{equation}
+\langle A(t) \rangle_x = \int dy \,A(y)\, p(y,t;x,0) = \langle -|A|  \psi(t) \rangle_x = \langle -|A e ^{-tH}| x\rangle
+\end{equation}
+\item
+\underline{Since $[H,K^0] = 0$} there exist a basis to study the stochastic process associated to
+$H$ where \underline{$K^0$ is diagonal}. We might consider the following representation
+\begin{eqnarray}
+\label{Koper2}
+K^+_i|\xi\rangle &=& \left(\frac{1}{2} + \xi\right)  |\xi+1\rangle\nonumber \\
+K^-_i|\xi\rangle &=& \xi  |\xi-1\rangle\nonumber \\
+K^o_i|\xi\rangle &=& \left(\xi + \frac{1}{4}\right)  |\xi\rangle
+\end{eqnarray}
+where $|\xi\rangle$ denotes a vector which together with its transposed
+$\langle \xi|$ form a complete basis of a Hilbert space and its dual, that is
+\begin{equation}
+\langle \xi|\eta \rangle = \delta_{\xi,\eta}
+\end{equation}
+If we use this representation then
+$$
+H = -{\cal L}^*
+$$
+and the probability mass function for the $\Xi(t)$ process is encoded in
+the state $|\phi(t)\rangle$, namely
+\begin{equation}
+|\phi(t) \rangle = \sum_{\xi} P(\xi,t) |\xi\rangle
+\end{equation}
+It immediately follows that
+\begin{equation}
+\langle \xi|\phi(t) \rangle = P(\xi,t)
+\end{equation}
+To compute expectation with respect to the $\Xi(t)$ process
+we introduce the flat state
+\begin{equation}
+\langle -_{dual} | = \sum_{\xi} \;\langle \xi|
+\end{equation}
+which is such that
+\begin{equation}
+\langle -_{dual} | \xi\rangle = 1
+\end{equation}
+Then for any observable $A=A(\Xi(t))$ we have that its expectation value
+at time $t$ can be written as
+\begin{equation}
+\langle A(t) \rangle_\xi = \sum_{\eta}\,A(\eta)\, p(\eta,t;\xi,0) = \langle -_{dual}|A|  \phi(t) \rangle_{\xi} = \langle -_{dual}|A e ^{-tH}| \xi\rangle
+\end{equation}
+\item
+\underline{The claim is the following: Duality, in general, is going from the basis
+where}\\
+\underline{one generator of the group is diagonal to a basis where another generator of}\\
+\underline{ the group is diagonal.}
+
+In our case we change from a basis where $K^+$ is diagonal to the base where $K^0$ is diagonal.
+
+\begin{eqnarray}
+\langle - |\prod_i\frac{(2K_i^+)^{\xi_i}}{(2\xi_i-1)!!}|\psi(t)\rangle_x
+& = &
+\int dy \; \langle y |\prod_i\frac{(2K_i^+)^{\xi_i}}{(2\xi_i-1)!!} e^{tL^*}|x\rangle \nonumber \\
+& = &
+\sum_{\eta} \int dy \; \langle y |\prod_i\frac{(2K_i^+)^{\eta_i}}{(2\eta_i-1)!!}e^{tL^*}|x\rangle \langle \eta|\xi\rangle\nonumber \\
+& = &
+\sum_{\eta} \int dy \; \langle y| \otimes \langle \eta| \prod_i\frac{(2K_i^+)^{\eta_i}}{(2\eta_i-1)!!} e^{tL^*} | x\rangle \otimes|\xi\rangle\nonumber \\
+& = &
+\sum_{\eta} \int dy \; \langle x| \otimes \langle \xi | e^{tL} \prod_i\frac{(2K_i^+)^{\eta_i}}{(2\eta_i-1)!!} | y\rangle \otimes|\eta\rangle \nonumber \\
+& = &
+\sum_{\eta} \int dy \; \langle x| \otimes \langle \xi | e^{tL} \prod_i\frac{y^{2\eta_i}}{(2\eta_i-1)!!} | y\rangle \otimes|\eta \rangle \nonumber \\
+& = &
+\sum_{\eta} \int dy \; \langle x| \otimes \langle \xi | e^{t{\cal L}} \prod_i\frac{y_i^{2\eta_i}}{(2\eta_i-1)!!} | y\rangle \otimes|\eta \rangle \nonumber \\
+& = &
+\sum_{\eta} \int dy \; \langle x| \otimes \langle \xi | e^{t{\cal L}} \prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!} | y\rangle \otimes|\eta \rangle \nonumber \\
+& = &
+\sum_{\eta} \int dy \; \langle y| \otimes \langle \eta |\prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!} e^{t{\cal L}^*}  | x\rangle \otimes|\xi \rangle\nonumber \\
+& = &
+\sum_{\eta} \int dy \; \langle \eta | \prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!} e^{t{\cal L}^*}  |\xi \rangle   \langle y | x\rangle   \nonumber \\
+& = &
+\sum_{\eta} \langle \eta | \prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!} |\phi(t)\rangle_{\xi}  \nonumber \\
+& = &
+\langle -_{dual} |\prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!}|\phi(t)\rangle_{\xi}
+\end{eqnarray}
+
+
+\end{itemize}
+
+\section{General k}
+
+A convenient $(2j+1)$-dimensional representation of the SU(2) algebra is given by
+\begin{eqnarray}
+J^+_i |n_i\rangle &=& (2j-n_i) |n_i+1\rangle \nonumber \\
+J^-_i |n_i\rangle &=& n_i      |n_i-1\rangle \nonumber \\
+J^0_i |n_i\rangle &=& (n_i-j)  |n_i\rangle
+\end{eqnarray}
+where the quantum numbers $n_i\in\{0,1,\ldots,2j\}$.
+{\bf Note that in this representation the adjoint of $J^+_i$ is not
+$J^-_i$, UNLESS $j=1/2$}.
+
+A matrix representation is:
+$$
+J^+ = \left(
+\begin{array}{cccc}
+ 0  &        &        &   \\
+ 2j & \ddots &        &   \\
+    & \ddots & \ddots &   \\
+    &        &    1   &  0\\
+\end{array}\right)
+\qquad
+J^- = \left(
+\begin{array}{cccc}
+ 0  &  1     &        &     \\
+    & \ddots & \ddots &     \\
+    &        & \ddots &  2j \\
+    &        &        &  0  \\
+\end{array}\right)
+\qquad
+J^0 = \left(
+\begin{array}{cccc}
+ -j  &        &        &   \\
+     & \ddots &        &   \\
+     &        & \ddots &   \\
+     &        &        &  j\\
+\end{array}\right)
+$$
+
+In the SU(1,1) case one can use the infinite dimensional representation
+\begin{eqnarray}
+\label{newrepresentationsu11}
+K^+_i |n_i\rangle &=& (2k+n_i) |n_i+1\rangle \nonumber \\
+K^-_i |n_i\rangle &=& n_i      |n_i-1\rangle \nonumber \\
+K^0_i |n_i\rangle &=& (n_i+k)  |n_i\rangle
+\end{eqnarray}
+where the quantum numbers $n_i\in\{0,1,2,\ldots\}$.
+A matrix representation is:
+$$
+K^+ = \left(
+\begin{array}{cccc}
+ 0  &        &        &   \\
+ 2k & \ddots &        &   \\
+    & 2k+1   & \ddots &   \\
+    &        & \ddots & \ddots\\
+\end{array}\right)
+\qquad
+K^- = \left(
+\begin{array}{cccc}
+ 0  &  1     &        &     \\
+    & \ddots &   2    &     \\
+    &        & \ddots &  \ddots \\
+    &        &        &  \ddots \\
+\end{array}\right)
+\qquad
+K^0 = \left(
+\begin{array}{cccc}
+ k   &        &        &   \\
+     &  k+1   &        &   \\
+     &        & k+2    &   \\
+     &        &        &  \ddots\\
+\end{array}\right)
+$$
+Let's check that in this representation the operator is stochastic.
+I will do it for the bulk:
+\begin{eqnarray}
+L_{i,i+1}|n_i,n_{i+1}\rangle
+&=&
+(2k+n_i) n_{i+1}|n_i +1 ,n_{i+1}-1\rangle \nonumber\\
+&+&
+n_i(2k+n_{i+1})|n_i -1 ,n_{i+1}+1\rangle \nonumber\\
+&+&
+(-2(n_i+k)(n_{i+1}+k)+2k^2)|n_i,n_{i+1}\rangle
+\end{eqnarray}
+The sum of the rates is
+$$
+(2k+n_i) n_{i+1}+
+n_i(2k+n_{i+1})
+-2(n_i+k)(n_{i+1}+k)+2k^2 =0
+$$
+
+
+
+
+
+
+
+
+
+% \end{document}
-- 
cgit v1.3


From 57c597f34858eb256a1c681ab931cf0ec665cec3 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 30 Dec 2020 13:49:20 +0100
Subject: Letter format and bibliography updates.

---
 bezout.bib |   17 +-
 bezout.tex |    2 +-
 cover.tex  | 1419 ++++--------------------------------------------------------
 3 files changed, 87 insertions(+), 1351 deletions(-)

(limited to 'cover.tex')

diff --git a/bezout.bib b/bezout.bib
index 8688ed2..9e600f0 100644
--- a/bezout.bib
+++ b/bezout.bib
@@ -267,6 +267,16 @@
  doi = {10.1007/jhep11(2012)023}
 }
 
+@book{Mezard_2009_Information,
+ author = {Mézard, Marc and Montanari, Andrea},
+ title = {Information, physics, and computation},
+ publisher = {Oxford University Press},
+ year = {2009},
+ address = {Great Clarendon Street, Oxford},
+ isbn = {9780198570837},
+ series = {Oxford Graduate Texts}
+}
+
 @article{Nguyen_2014_The,
  author = {Nguyen, Hoi H. and O'Rourke, Sean},
  title = {The Elliptic Law},
@@ -294,12 +304,7 @@
  url = {https://doi.org/10.2307%2F2371510},
  doi = {10.2307/2371510}
 }
-@book{mezard2009information,
-  title={Information, physics, and computation},
-  author={Mezard, Marc and Montanari, Andrea},
-  year={2009},
-  publisher={Oxford University Press}
-}
+
 @inproceedings{Scorzato_2016_The,
  author = {Scorzato, Luigi},
  title = {The {Lefschetz} thimble and the sign problem},
diff --git a/bezout.tex b/bezout.tex
index 96ad257..d7f4dc7 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -44,7 +44,7 @@
 
 Spin-glasses have long been considered the paradigm of many variable `complex
 landscapes,' a subject that includes neural networks and optimization problems,
-most notably constraint satisfaction \cite{mezard2009information}.  The most tractable family of these
+most notably constraint satisfaction \cite{Mezard_2009_Information}.  The most tractable family of these
 are the mean-field spherical $p$-spin models \cite{Crisanti_1992_The} (for a
 review see \cite{Castellani_2005_Spin-glass}) defined by the energy
 \begin{equation} \label{eq:bare.hamiltonian}
diff --git a/cover.tex b/cover.tex
index 586fa38..2d2fc10 100644
--- a/cover.tex
+++ b/cover.tex
@@ -1,1358 +1,89 @@
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+\usepackage[utf8]{inputenc} % why not type "Bézout" with unicode?
+\usepackage[T1]{fontenc} % vector fonts plz
+\usepackage{newtxtext,newtxmath} % Times for PR
+\usepackage[
+  colorlinks=true,
+  urlcolor=purple,
+  citecolor=purple,
+  filecolor=purple,
+  linkcolor=purple
+]{hyperref} % ref and cite links with pretty colors
+\usepackage{xcolor}
+
+\signature{
+  Jaron Kent-Dobias \& Jorge Kurchan
 }
-% 
 
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+\address{
+  Laboratoire de Physique \\
+  Ecole Normale Sup\'erieure \\
+  24, rue Lhomond \\ 
+  75005 Paris
 }
 
-\setstretch{1.24}
-
 \begin{document}
-
-\title{{\bf COVER LETTER \\`Complex complex landscapes'}}
-%\footnote{{\bf Key-words}: }
-%}}
-
-\author{
-Jaron Kent-Dobias
- and 
-Jorge Kurchan
+\begin{letter}{
+  Editorial Office\\
+  Physical Review Letters\\
+  1 Research Road\\
+  Ridge, NY 11961
 }
 
-\maketitle
-
-
-
-\vspace{1.cm}
-
-
-The subject of `Complex Landscapes', which started in the spin-glass literature, is concerned with functions (landscapes) of many variables, having a multiplicity of minimums, which are the objects of interest. Apart from its obvious interest for glassy systems, it has found a myriad applications  in  many domains:  Computer Science, Ecology, Economics, Biology \cite{mezard2009information}.
-
-In the last few years, a renewed interest has developed for landscapes for which the variables are complex. There are a few reasons for this: {\em i)} in Computational Physics, there is the main obstacle of the `sign problem', and a strategy has emerged to attack it deforming the sampling space into complex variables. This is a most natural and promising path, and any progress made will have game-changing impact in solid state physics and lattice-QCD \cite{Cristoforetti_2012_New,Scorzato_2016_The}.
-{\em ii)} At a more basic level, following the seminal work of E. Witten \cite{Witten_2010_A,Witten_2011_Analytic}, there has been a flurry of activity concerning the very definition of quantum mechanics, which requires also that one move into the complex plane. 
-
-In all these cases, just like in the real case, one needs to know the structure of the `landscape', where are the saddle points and how they are connected, typical questions of `complexity'.
-However, to the best of our knowledge, there are no studies extending the methods of the theory of
-complexity to
-complex variables. 
-We believe our paper will open a field that may find 
-numerous applications and will widen our theoretical view of complexity in general.
-
-
+\opening{}
+
+The subject of `Complex Landscapes,' which started in the spin-glass
+literature, is concerned with functions (landscapes) of many variables, having
+a multiplicity of minimums, which are the objects of interest. Apart from its
+obvious interest for glassy systems, it has found a myriad applications  in
+many domains:  Computer Science, Ecology, Economics, Biology
+\cite{Mezard_2009_Information}.
+
+In the last few years, a renewed interest has developed for landscapes for
+which the variables are complex. There are a few reasons for this: {\em i)} in
+Computational Physics, there is the main obstacle of the `sign problem', and a
+strategy has emerged to attack it deforming the sampling space into complex
+variables. This is a most natural and promising path, and any progress made
+will have game-changing impact in solid state physics and lattice-QCD
+\cite{Cristoforetti_2012_New,Scorzato_2016_The}.  {\em ii)} At a more basic
+level, following the seminal work of E. Witten
+\cite{Witten_2010_A,Witten_2011_Analytic}, there has been a flurry of activity
+concerning the very definition of quantum mechanics, which requires also that
+one move into the complex plane. 
+
+In all these cases, just like in the real case, one needs to know the structure
+of the `landscape', where are the saddle points and how they are connected,
+typical questions of `complexity'.  However, to the best of our knowledge,
+there are no studies extending the methods of the theory of complexity to
+complex variables.  We believe our paper will open a field that may find
+numerous applications and will widen our theoretical view of complexity in
+general.
+
+\closing{Sincerely,}
+\end{letter}
+
+\bibliographystyle{unsrt}
 \bibliography{bezout}
 
-
 \end{document}
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-\section{The Kipnis-Marchioro-Presutti model}
-
-Consider the following process: 
-\begin{itemize}
-\item
-choose a pair of neighbouring sites and completely
-exchange energy between them
-\item
-if the site is one of the borders, exchange completely energy with the bath.
-\end{itemize}
-each choice with probability $1/(N+1)$. From here onwards, we shall denote
-$\tau$ a large time, sufficient for any two-site thermalisation.
-
-The evolution operator in one step is:
-\begin{eqnarray}
-U &=& \frac{1}{N+1} \left[ e^{-\tau L_1^*} +  e^{-\tau L_N^*} + \sum_{i=1}^{N-1}   e^{-\tau L^*_{i,i+1}} \right]
-\nonumber \\
-&=& \frac{1}{N+1} \left[ e^{-2\tau (T_1 K^-_1 + K^o_1 + k) } +  e^{-2\tau(T_L K^-_L + K^o_L +k)  }
- + \sum_{i=1}^{N-1}   e^{ \frac{-\tau}{k}
-(K^+_i K^-_{i+1} + K^-_i K^+_{i+1} - 2 K^o_i K^o_{i+1}
-+ 2k^2 )} \right] \nonumber \\
-~
-\end{eqnarray}
-and the dynamics after $n$ steps is given by $U^n$.
-Because we are considering large $\tau$, the terms in the sums are  in fact projectors
-onto the lowest eigenvalues of the exponents. We shall however keep the notation as it is
-in order to stress the symmetry of the bulk terms.
-
-Let us now show that - at the level of energies - this dynamics yields the KMP process 
-{\em for $k=\frac{1}{2}$, that is $m=2$}.
-Consider first a general $m$, and two neighbouring sites of coordinates $x = \{x_\alpha\}_{\alpha=1,\ldots,m}$, 
-$y=\{y_\alpha\}_{\alpha=1,\ldots,m}$.
-If they are completely thermalised, it means that (cfr (\ref{bb}):
-the joint probability density satisfies
-\begin{equation}
-\left(x_{\alpha}
-\frac{\partial}{\partial y_{\beta}} -
-y_{\beta}\frac{\partial}{\partial x_{\alpha}}
-  \right) p(x,y)=0
-\end{equation}
-It is easy to see that this may happen if and only if
-\begin{equation}
-p(x,y)= p[ \sum_\alpha (x_\alpha^2+y_\alpha^2)]
-\end{equation}
-In particular let us consider the microcanonical measure
-\begin{equation}
-p(x,y)= \delta[ \sum_\alpha (x_\alpha^2+y_\alpha^2)-\epsilon ]
-\end{equation}
-Defining new random variables $\epsilon_1$ and $\epsilon_2$ 
-as the energies of the neighboring sites
-\be
-\epsilon_1 = \sum_\alpha x_\alpha^2
-\ee
-\be
-\epsilon_2 = \sum_\alpha y_\alpha^2
-\ee
-then their joint probability density will be
-\begin{equation}
-p(\epsilon_1,\epsilon_2) = \frac{S_m^2}{4} \delta(\epsilon_1+\epsilon_2-\epsilon)
-\epsilon_1^{\frac{1}{2}-1} \epsilon_2^{\frac{1}{2}-1}
-\end{equation}
-where $S_m$ denotes the surface of the unit sphere in $m$ dimension
-\be
-S_m = \frac{m \pi^{m/2}}{\Gamma(\frac{1}{2}+1)}
-\ee
-{\em This yields a flat distribution for $m=2$, i.e. the KMP model.}
-
-
-
-
-\section{Dual model}
-
-
-The expectation value of an observable at time $t$, starting from an initial
-distribution $|init\rangle$ is:
-
-
-\begin{equation}
-<O> = \langle - | O e^{-Ht} | init \rangle
-\end{equation}
-where $\langle - |$ is a constant.
-Taking the adjoint $ x_i \to x_i$, $\partial_i \to -\partial_i$:
-\begin{equation}
-<O> = \langle - | O e^{-Ht} | init \rangle= \langle init|  e^{-H^\dag t} O |- \rangle
-\end{equation}
-where $H^\dag(K^\pm, K^o)=H( K^\pm, -K^o)$ (because of the change of signs of the derivatives)
-\begin{eqnarray}
--H^\dag&=& \frac{4}{1} \sum_i \left(
-K^+_i K^-_{i+1} + K^-_i K^+_{i+1} - 2 K^o_i K^o_{i+1}
-+ \frac{m^2}{8} \right)
-\nonumber\\
-&+&2 \left(T_1 K^-_1 + K^o_1 + \frac{1}{4}\right)
-+2 \left(T_L K^-_L + K^o_L +\frac{1}{4}\right)
-\end{eqnarray}
-In particular, for the generating function we had chosen
- \begin{equation}
- O |- \rangle = \Pi_i \frac{x_i^{2 \xi_i}}{(2\xi_i -1)!!}|-\rangle=|\xi_1,...,\xi_N\rangle
-\end{equation}
-
-Considered as an operator acting on `particle number', as counted by $K^o$, $H^\dag$ does not
-conserve the probability.
-The trick we used can be expressed as follows: introduce the particle number $\xi_o$ and $\xi_{N+1}$ 
-and the operators $A^+_o$ and $A^+_{N+1}$, which create particles in boundary sites with unit rate.
-We consider now the {\em enlarged} process generated by
-\begin{eqnarray}
--H^{dual}&=& \frac{4}{1} \sum_i \left(
-K^+_i K^-_{i+1} + K^-_i K^+_{i+1} - 2 K^o_i K^o_{i+1}
-+ \frac{m^2}{8} \right)
-\nonumber\\
-&+&2 \left(A^+_o K^-_1 + K^o_1 - \frac{1}{4}\right)
-+2 \left( A^+_{N+1} K^-_N + K^o_N -\frac{1}{4}\right)
-\end{eqnarray}
-which conserves ({\it seems}) particle number and probability.
-We wish to prove that:
-
-\begin{eqnarray}
-<O> &=& \langle init| e^{-H^\dag t} |\xi_1,...,\xi_N \rangle \nonumber \\
-&=& \sum_{\xi_o,\xi_{N+1}}
-  T_1^{\xi_o} T_{L}^{\xi_{N+1}} \langle \xi_o \xi_{N+1} | \otimes \langle
-  init| e^{-H^{dual} t} |\xi_1,...,\xi_N \rangle \otimes
-  |\xi_o=0,\xi_{N+1}=0 \rangle \nonumber \\
-\label{ggg}
-\end{eqnarray}
-
-
-I think the proof is obvious, because developing the exponential of $H^{dual}$ all the $A^+$ can be
-collected because they commute with everything else, and the experctation value
-\begin{equation}
-\sum_{\xi_o}  T_1^{\xi_o}  \langle \xi_o  |[A^+_o]^r  |\xi_o=0 \rangle = T_1^r
- \end{equation}
-just puts back as many $T$'s as necessary.
-
-I do not know exactly how to use (\ref{ggg}) in general, but in the large time limit the evolution
-voids the chain of particles
-
-
-\section{Dual of KMP}
-
-I think that the argument runs through without changes if we use $U$ defined for the KMP model.
-We just have to note that each term corresponds to an evolution of two sites (or a site and the bath)
-and so in the dual it corresponds to sharing the particles between those two sites, or emptying
-the sites at the borders.
-
-{\bf: NOTE by Cristian}
-
-We can check that the duality function chosen in the original paper by KMP
-do coincide with the duality function of our process for $m=2$ (and the random
-variables are the energies).
-Indeed we start from
-\be
-f(x,\xi) = \prod_i (\sum_{\alpha} x_{i,\alpha}^2)^{\xi}
-\ee
-When the bath have equal temperature (let's us choose T=1) then the stationary 
-measure is 
-\be
-\pi(x) = \prod_i \frac{1}{(2\pi)^{m/2}} \exp\left(-\sum_{\alpha}\frac{x_{i,\alpha}^2}{2}\right)
-\ee
-Let us focus on a fixed $i$ (that is in this short computation we write $x$ for $x_i$). 
-We have
-\begin{eqnarray}
-\E(f(x,\xi)) 
-&=&
-\int dx_1 \cdots \int dx_m (x_1^2+\ldots + x_m^2)^{\xi} \exp-\left(\frac{x_{1}^2}{2}+\ldots+\frac{x_{1}^2}{2}\right) 
-\nonumber \\
-& = & 
-\int dr S_m r^{2\xi} \exp-\left(\frac{r^2}{2}\right)
-\nonumber \\
-& = &
-\frac{\frac{1}{2}\Gamma(\frac{1}{2}+\xi)}{\Gamma(\frac{m}{2}+1)} 2^\xi
-\nonumber \\
-\end{eqnarray}
-Special cases:
-\begin{itemize}
-\item $m=1$
-
-$$
-\E(f(x,\xi)) = (2\xi-1)!!
-$$
-where one uses that $\Gamma(\frac{1}{2}+\xi)= \frac{\sqrt{\pi}(2\xi-1)!!}{2^{\xi}}$ and $\Gamma(\frac{3}{2}) = \frac{\sqrt{\pi}}{2}$
-\item $m=2$
-
-$$
-\E(f(x,\xi)) = \xi! 2^\xi
-$$
-where one uses that $\Gamma(1+\xi)= \xi!$ and $\Gamma(2) = 1$.
-Thus, if one defines the energies as 
-$$
-\epsilon_i = \sum_{\alpha}\frac{x_{i,\alpha}^2}{2}
-$$
-one recover the choice of KMP for the dual function
-$$
-O(\epsilon_i,\xi) = \prod_i \frac{\epsilon_i^{\xi_i}}{\xi_i!}
-$$
-\end{itemize}
-
-
-
-
-
-
-\section{ Dual of SEP: here goes an outline of how to proceed for the SSEP}
-
-
-\be
-H=-L_{SEP}^*
-\ee
-\begin{eqnarray}
-L^*_{SEP} &=& \frac{1}{j}
-  \sum_i  \left(J^+_i J^-_{i+1} + J^-_i J^+_{i+1} + 2 J^o_i J^o_{i+1}
- - 2 j^2  \right)\\
-&+&\alpha (J^-_1 - J^o_1-j) + \gamma (J^+_1 + J^o_1-j)
-+ \delta (J^-_L - J^o_L-j) + \beta (J^+_L + J^o_L-j)\nonumber
-\end{eqnarray}
-The factor $1/j$ is analogous to the factor $1/m$ in (\ref{bb}).
-The operators $J^+_i, J^-_i, J^o_i$ act on the Hilbert space
- corresponding to  $0 \le r \le n$  particles per site $\otimes_i |r\rangle_i$
-as follows:
-\begin{eqnarray}
-J^+_i |r\rangle_i &=& (2j-r)  |r+1\rangle_i \nonumber \\
- J^-_i |r\rangle_i &=& r  |r-1\rangle_i \nonumber \\
-J^o_i |r\rangle_i &=& (r-j)  |r\rangle_i
-\end{eqnarray}
-
-The conjugation properies are as follows. There is an operator $Q$,
-{\em diagonal in this basis  } (I give the expression below), such that:
-\begin{equation}
-[J^+_i]^\dag = Q[J^-_i]Q^{-1} \qquad [J^-_i]^\dag = Q[J^+_i]Q^{-1}
-\end{equation}
-while $[J^z_i]^\dag=J^z_i= Q[J^z_i]Q^{-1}$.
-
-
-
-The expectation value of an observable at time $t$, starting from an initial
-distribution $|init\rangle$ is:
-
-
-\begin{equation}
-<O> = \langle - | O e^{-Ht} | init \rangle
-\end{equation}
-where $\langle - |$ is a constant.
-As before:
-\begin{eqnarray}
-<O> &=& \langle - | O e^{-Ht} | init \rangle=
-\langle init|  e^{-H^\dag t} O |- \rangle= \nonumber \\
-& & \langle init|Q  e^{-{\bar H} t}  Q^{-1}O   |- \rangle=
-\langle init|Q \;  e^{-{\bar H} t}  Q^{-1}O Q  Q^{-1} |- \rangle
-\end{eqnarray}
-
-
-{\em  $ {\bar H}$ is the same operator as $H$ but with
-$J^+$ substituted by  $J^-$, and vice-versa.}
-Our job is now to make the rotation that will eliminate the $J^+$'s in
-the border terms of $ {\bar H}$.
-
-
-
-
-The transformation is of the form
-\begin{eqnarray}
-e^{\mu J^+} J^+ e^{-\mu J^+}&=&J^+ \nonumber \\
-e^{\mu J^+} J^o e^{-\mu J^+} &=&J^o - \mu J^+ \nonumber \\
-e^{\mu J^+} J^- e^{-\mu J^+} &=& J^- + 2 \mu J^o - \mu^2 J^+
-\end{eqnarray}
-for suitable $\mu$.
-Putting $\mu=-1$, we get that {\bf the bulk term is left invariant,
-precisely because of the SU(2) symmetry}. The boundary terms {\bf of $\bar H$}
-transform further into:
-\begin{eqnarray}
-& e^{\mu J^+_1} \left[ \alpha (J^+_1 - J^o_1-j) + \gamma (J^-_1 + J^o_1-j)
-\right] e^{-\mu J^+_1}= \nonumber \\ & \gamma(J^-_1 + 2 \mu J^o_1 - \mu^2
-J^+_1 +J^o_1 - \mu J^+_1 -j) + \alpha (J^+_1 - J^o_1 + \mu J^+_1 -j)
-= \nonumber \\
-& \alpha(- J^o_1 -j) + \gamma (J^-_1 -J^o_1 -j)
-\label{trans}
-\end{eqnarray}
-which is of the same form we have in the $SU(1,1)$ model.
-The same can be done in the other boundary term.
-
-We thus get:
-\begin{eqnarray}
-<O> &=& \langle - | O e^{-Ht} | init \rangle=
-\langle init|Q \; e^{-{\bar H} t}  Q^{-1}O Q  Q^{-1} |- \rangle \nonumber\\
-&= & \langle init|Q   e^{ \sum_i J^+_i}  e^{-{\bar H_{dual}} t}
- e^{ -\sum_i J^+_i}  Q^{-1}O Q  Q^{-1} |- \rangle \nonumber \\
-&= & \langle init|Q   e^{ \sum_i J^+_i}  e^{-{\bar H_{dual}} t}
- e^{ -\sum_i J^+_i}  Q^{-1}O Q e^{ \sum_i J^+_i}
-  e^{ -\sum_i J^+_i}  |- \rangle \nonumber \\
- &= & \langle init|Q  Q^{-1}  e^{ \sum_i J^+_i}  e^{-{\bar H_{dual}} t}
- e^{ -\sum_i J^+_i}  Q^{-1}O Q e^{ \sum_i J^+_i}  |-_{dual} \rangle
-\end{eqnarray}
-where we have defined $H_{dual}$  as the transformed Hamiltonian.
-
-We now have to study $ |-_{dual} \rangle \equiv e^{ -\sum_i J^+_i}
- Q^{-1} |- \rangle$
-Because we know that  terms like those proportional to $\gamma$ and $\alpha$
-anihilate the measure to the left:
-\begin{eqnarray}
-& & \langle - | (J^-_i - J^o_i-j) =0\nonumber \\
-& & \langle - | (J^+_i + J^o_i-j) =0
-\end{eqnarray}
-this implies that in the new variables and following all the transformations
-(cfr (\ref{trans})):
-\begin{eqnarray}
-& & (J^-_i -J^o_i -j)e^{ -\sum_i J^+_i} Q^{-1} |- \rangle= 0 \nonumber \\
-& & ( -J^o_i -j)e^{ -\sum_i J^+_i} Q^{-1} |- \rangle =0
-\end{eqnarray}
-which implies that $( J^o_i +j) |-_{dual} \rangle= J^-_i  |-_{dual} \rangle=0$,
-and this means that
-\begin{equation}
-J^o_i |-_{dual} \rangle =-j |-_{dual}  \rangle
-\end{equation}
-is the vacuum of particles in this base!
-
-All in all we are left with:
-\begin{eqnarray}
-<O> &=&  \langle init|Q \;  e^{ \sum_i J^+_i}
-  e^{-{\bar H_{dual}} t} e^{ -\sum_i J^+_i}  Q^{-1}O Q e^{ \sum_i J^+_i}
-  |-_{dual} \rangle \nonumber \\
- &=&  \langle init|Q \;  e^{ \sum_i J^+_i}
-  e^{-{\bar H_{dual}} t} {\hat O}
-  |-_{dual} \rangle
-\end{eqnarray}
-where $ {\hat O} \equiv e^{ -\sum_i J^+_i} Q^{-1}O Q e^{ \sum_i
-J^+_i}$.  We have to start with the
-vacuum $ |-_{dual} \rangle$, then apply $ {\hat O} $, (which creates
-particles because it contains many $J^+$'s), and then there is the
-dual evolution. The final configuration has to be overlapped with
-$\langle f| \equiv  \langle init|Q \;  e^{ \sum_i J^+_i}$.
-For large times, there will be no particle left except in the two extra sites
-in the borders.
-
-\section{Constructive approach}
-
-Here I would like to say the following: if I have a modle of transport
-of which I do not know if it has a Dual one, I can proceed as follows.
-I take a small version with no baths and a few sites. I write the
-evolution operator and I diagonalise it numerically.  If there is a
-non abelian group, the eigenvalues will be in degenerate
-multiplets. Hence, if I find multiplets, then very probably there is a
-dual model, if I do not, then there cannot be one.  It would be nice
-to show it with the KMP model with two or three sites.
-
-Another thing is to consider higher groups. $SU(3)$ has already been studied
-for two kinds of particles. We know how to map to a dual in that
-case, if it has not been done yet.
-
-\newpage
-{\bf THIS PART HAS BEEN WRITTEN BY CRISTIAN} 
-
-The aim of this file is to set notation in the two languages.
-Let us focus on duality for the case we already know:
-SU(1,1) model with $k=1/4$. To fix ideas let us consider only
-the bulk part of the system with periodic boundary conditions.
-
-\section{Probabilistic language}
-We have two stochastic Markovian process with continuous time.
-\begin{itemize}
-\item
-\underline{The first process $X(t) \in \R^N$} is given by the Fokker-Planck equation:
-\be
-\frac{dp(x,t)}{dt}  = L^* p(x,t)
-\ee
-where $p(x,t)$ represents the probability density
-for the process $X(t)$, that is
-$$
-p(x,t)dx = Prob (X(t)\in (x,x+dx))
-$$
-and
-\begin{eqnarray}
-L^*p(x,t)
-& = &
-\sum_i L^*_{i,i+1} p(x,t) \noindent\\
-& = &
-\sum_i \left(x_i\frac{\partial}{\partial x_{i+1}} -x_{i+1}\frac{\partial}{\partial x_{i}}\right)^2 p(x,t)
-\end{eqnarray}
-\item
-\underline{The second process $\Xi(t) \in \N^N$} is characterized by the master equation
-\be
-\frac{dP(\xi,t)}{dt} = {\cal L^*} P(\xi,t)
-\ee
-where $P(\xi,t)$ represents the
-probability mass function for the process $\Xi(t)$, that is
-$$
-P(\xi,t) = Prob (\Xi(t) = \xi)
-$$
-and
-\begin{eqnarray}
-{\cal L}^*P(\xi,t)
-& = &
-\sum_i {\cal L}^*_{i,i+1}P(\xi,t) \nonumber \\
-& = &
-\sum_i 2\xi_i \left(1+ 2\xi_{i+1}\right) P(\xi^{i,i+1},t)
-+ \left(1+2\xi_i\right)2\xi_{i+1} P(\xi^{i+1,i},t) \nonumber\\
-& & - 2\left(2\xi_i + \frac{1}{2}\right)\left(2\xi_{i+1} + \frac{1}{2}\right) P(\xi,t)
-+ \frac{1}{2}P(\xi,t)
-\end{eqnarray}
-and $\xi^{i,j}$ denotes the configuration that is obtained by removing one particle
-at $i$ and adding one particle at $j$.
-\newpage
-\item
-\underline{In general, Duality means the following}:
-there exists functions $O(x,\xi): \R^N \times \N^N \mapsto \R$ such that
-the following equality between expectations for the two processes holds
-\begin{center}
-\fbox{\parbox{9cm}{
-\be
-\E_x( O(X(t),\xi)) =\E_\xi(O(x,\Xi(t)))
-\ee
-}}
-\end{center}
-The subscripts in the expectations denote the initial conditions of the processes:
-$X(0) =x$ on the left and $\Xi(0) = \xi$ on the right.
-More explicitly we have:
-\be
-\int dy O(y,\xi) p(y,t; x,0)  = \sum_{\eta} O(x,\eta) P(\eta,t; \xi,0)
-\ee
-To prove duality it is sufficient to show that
-\be
-\label{main}
-L O(x,\xi) = {\cal L} O(x,\xi)
-\ee
-where $L$, that is working on $x$, is the adjoint of  $L^*$ and ${\cal L}$, that is working on $\xi$,
-is the adjoint of  ${\cal L}^*$.
-Indeed we have:
-\begin{eqnarray}
-\E_x( O(X(t),\xi))
-& = &
-\int dy O(y,\xi) p(y,t; x,0) \\
-& = &
-\sum_{\eta} \int dy O(y,\eta) p(y,t; x,0) \delta_{\eta,\xi} \\
-& = &
-\sum_{\eta} \int dy O(y,\eta) e^{tL^*} \delta(y-x) \delta_{\eta,\xi} \\
-& = &
-\sum_{\eta} \int dy e^{tL} O(y,\eta)  \delta(y-x) \delta_{\eta,\xi} \\
-& = &
-\sum_{\eta} \int dy e^{t{\cal L}} O(y,\eta)  \delta(y-x) \delta_{\eta,\xi}  \\
-& = &
-\sum_{\eta} \int dy O(y,\eta) e^{t{\cal L}^*} \delta(y-x) \delta_{\eta,\xi}  \\
-& = &
-\sum_{\eta} \int dy O(y,\eta) P(\eta,t;\xi,0) \delta(y-x) \\
-& = &
-\sum_{\eta}  O(x,\eta) P(\eta,t;\xi,0)  \\
-& = &
-\E_\xi(O(x,\Xi(t)))
-\end{eqnarray}
-\newpage
-\item
-\underline{For the present case, the proper function to be considered are}
-\be
-\label{Oss}
-O(x,\xi) = \prod_{i} \frac{x_i^{2\xi_i}}{(2\xi_i-1)!!}
-\ee
-Let us check Eq.(\ref{main}) on this choice. We have
-\begin{eqnarray*}
-&&
-L_{i,i+1} O(x,\xi)
-=
-\left(\prod_{k\not\in\{i,i+1\}}  \frac{x_k^{2\xi_k}}{(2\xi_k -1)!!}\right)
-\times
-\\
-&&\left(2\xi_{i+1}(2\xi_{i+1}-1) \frac{x_i^{2\xi_i+2}}{(2\xi_i -1)!!}\frac{x_{i+1}^{2\xi_{i+1}-2}}{(2\xi_{i+1} -1)!!}
-- 2\xi_{i}(2\xi_{i+1}+1) \frac{x_i^{2\xi_i}}{(2\xi_i -1)!!}\frac{x_{i+1}^{2\xi_{i+1}}}{(2\xi_{i+1} -1)!!}
-\right.
-\\
-&&\left.- 2\xi_{i+1}(2\xi_{i}+1) \frac{x_i^{2\xi_i}}{(2\xi_i -1)!!}\frac{x_{i+1}^{2\xi_{i+1}}}{(2\xi_{i+1} -1)!!}
-+2\xi_{i}(2\xi_{i}-1) \frac{x_i^{2\xi_i-2}}{(2\xi_i -1)!!}\frac{x_{i+1}^{2\xi_{i+1}+2}}{(2\xi_{i+1} -1)!!}
-\right)
-\\
-\end{eqnarray*}
-which implies
-\begin{eqnarray*}
-L_{i,i+1} O(x,\xi)
-& = &
-\Big(2\xi_{i+1}(2\xi_{i}+1) [O(x,\xi^{i+1,i})-O(x,\xi)]
-\\
-&&
-\;+\;2\xi_{i}(2\xi_{i+1}+1) [O(x,\xi^{i,i+1})-O(x,\xi)]\Big)
-\\
-& = &
-{\cal L}_{i,i+1} O(x,\xi)
-\end{eqnarray*}
-
-\item \underline{How to find the proper normalization?}
-Suppose that we are in the general following situation:
-\begin{itemize}
-\item We have a generator $L$ of a Markov process $X(t)$.
-\item We know its stationary measure $\pi(x)$:
-\be
-L^* \pi(x) = 0
-\ee
-\item We have functions $f(x,\xi)$ for which the following holds:
-\be
-\label{aaa}
-L f(x,\xi) = \sum_{\eta} r(\xi,\eta) f(x,\eta)
-\ee
-with
-\be
-\label{bbb}
-r(\xi,\eta) \ge 0 \qquad  \mbox{if}\quad \xi \neq \eta
-\ee
-\be
-\label{ccc}
-r(\xi,\xi)  \le 0 \qquad  \mbox{if}\quad \xi = \eta
-\ee
-\end{itemize}
-The matrix $r$ resembles the generator of a dual Markov process,
-but it is not because it does not satisfy the condition
-$\sum_{\eta} r(\xi,\eta) = 0$.
-In order to find the generator of the dual process we proceed as
-follows:
-\begin{enumerate}
-\item Define
-\be
-m(\xi) = \int f(x,\xi) \pi(x) dx
-\ee
-\item Define
-\be
-q(\xi,\eta)= m(\xi)^{-1} r(\xi,\eta) m(\eta)
-\ee
-\item Define
-\be
-O(x,\xi) = m(\xi)^{-1} f(x,\xi)
-\ee
-\end{enumerate}
-Then the matrix $q$ can be seen as the generator of the dual Markov process $\Xi(t)$, that is
-\be
-L O(x,\xi) = \sum_{\eta} q(\xi,\eta) O(x,\eta)
-\ee
-with
-\be
-q(\xi,\eta) \ge 0 \qquad  \mbox{if}\quad \xi \neq \eta
-\ee
-\be
-q(\xi,\xi)  \le 0 \qquad  \mbox{if}\quad \xi = \eta
-\ee
-\be
-\sum_{\eta} q(\xi,\eta) = 0
-\ee
-Indeed we have:
-\begin{eqnarray}
-L O(x,\xi)
-&=&
-L m(\xi)^{-1} f(x,\xi) \nonumber \\
-&=&
-m(\xi)^{-1} \sum_{\eta} r(\xi,\eta) f(x,\eta) \nonumber \\
-&=&
-m(\xi)^{-1} \sum_{\eta} m(\xi)q(\xi,\eta) m(\eta)^{-1} m(\eta) O(x,\eta)\nonumber \\
-&=&
-\sum_{\eta} q(\xi,\eta) O(x,\eta)
-\end{eqnarray}
-and
-\begin{eqnarray}
-\sum_{\eta} q(\xi,\eta)
-&=&
-\sum_{\eta} m(\xi)^{-1} r(\xi,\eta) m(\eta)  \nonumber \\
-&=&
-m(\xi)^{-1} \sum_{\eta}  r(\xi,\eta) \int f(x,\eta) \pi(x) dx \nonumber \\
-&=&
-m(\xi)^{-1} \int L f(x,\xi) \pi(x) dx \nonumber \\
-&=&
-m(\xi)^{-1} \int f(x,\xi) L^* \pi(x) dx \nonumber \\
-&=&
-0
-\end{eqnarray}
-
-
-
-\item \underline{Our case}. Among all the invariant measure
-of the $X(t)$ process, namely the normalized function with
-spherical symmetry $p(x) = p(\sum_i x_i^2)$, a special role is
-played by the Gibbs measure
-$$
-\pi(x)
-= \left(\frac{\beta}{2\pi}\right)^{(N/2)} e^{-\beta\sum_i \frac{x_i^2}{2}}
-= \left(\frac{\beta}{2\pi}\right)^{(N/2)} \prod_i e^{-\beta\frac{x_i^2}{2}}
-$$
-which is selected as soon as the system is placed in contact with
-thermal bath working at inverse temperature $\beta$.
-Moreover: If $Z$ is a centered Gaussian, namely $Z\sim N(0,\sigma^2)$,
-then
-$$
-\E(Z^{2n}) = \sigma^{2n} (2n-1)!!
-$$
-If one start from
-$$
-f(x,\xi) = \prod_i x_i^{2\xi}
-$$
-which satisfy (\ref{aaa}),(\ref{bbb}),(\ref{ccc}) and apply
-the previous procedure, one arrives to (\ref{Oss}).
-
-{\bf Remark:} Note that, in applying the procedure, the
-dependence on $\beta$ disappear!!!!
-\end{itemize}
-
-
-\section{Quantum language}
-
-
-Here we start from a quantum spin chain
-$$
-H = - 4 \sum_i \left( K^+_iK^-_{i+1} + K^-_iK^+_{i+1} -2 K^0_iK^0_{i+1} + \frac{1}{8}\right)
-$$
-where the spin $K_i$'s satisfy the SU(1,1) algebra
-\begin{eqnarray}
-\label{commutatorsSU11}
-[K_i^{0},K_i^{\pm}] &=& \pm K_i^{\pm} \nonumber \\
-{[}K_{i}^{-},K_{i}^{+}{]} &=& 2K_i^{0}
-\end{eqnarray}
-We are going to see the Schr\"odinger equation with imaginary time
-\begin{equation}
-\label{schroedinger}
-\frac{d}{dt}|\psi(t) \rangle = -H |\psi(t)\rangle\;.
-\end{equation}
-as the evolution equation for the probability distribution of
-a Markovian stochastic process.
-\begin{itemize}
-\item
-\underline{The Hamiltonian possesses the SU(1,1) invariance}. If we define
-\be
-K^+ = \sum_{i} K_i^+
-\ee
-\be
-K^- = \sum_{i} K_i^-
-\ee
-\be
-K^0 = \sum_{i} K_i^0
-\ee
-we find that
-\be
-[H,K^+] = 0
-\ee
-\be
-[H,K^-] = 0
-\ee
-\be
-[H,K^0] = 0
-\ee
-\item
-\underline{Since $[H,K^+] = 0$} there exist a basis to study the stochastic process associated to
-$H$ where \underline{$K^+$ is diagonal}. We might consider the following representation
-\begin{eqnarray}
-\label{Koper}
-K^+_i &=& \frac{1}{2}  x_{i}^2 \nonumber \\
-K^-_i &=& \frac{1}{2} \frac{\partial^2}{\partial x_{i}^2} \nonumber \\
-K^o_i &=& \frac{1}{4}  \left\{\frac{\partial}{\partial x_{i}} x_{i} +
- x_{i} \frac{\partial}{\partial x_{i}} \right \}
-\end{eqnarray}
-If we use this representation then
-$$
-H = -L^*
-$$
-and the probability density function for the $X(t)$ process is encoded in
-the state $|\psi(t)\rangle$, namely
-\begin{equation}
-|\psi(t) \rangle = \int dx p(x,t) |x\rangle
-\end{equation}
-where we have introduced the notation $|x\rangle$ to denote a completely
-localized state, that is a vector which together with its transposed
-$\langle x|$ form a complete basis of a Hilbert space and its dual:
-\begin{equation}
-\langle x|x' \rangle = \delta(x-x')
-\end{equation}
-It immediately follows that
-\begin{equation}
-\langle x|\psi(t) \rangle = p(x,t)
-\end{equation}
-To compute expectation with respect to the $X(t)$ process
-we introduce the flat state
-\begin{equation}
-\langle - | = \int dx \;\langle x|
-\end{equation}
-which is such that
-\begin{equation}
-\langle - | x\rangle = 1
-\end{equation}
-Then for any observable $A = A(X(t))$ we have that its expectation value
-at time $t$ can be written as
-\begin{equation}
-\langle A(t) \rangle_x = \int dy \,A(y)\, p(y,t;x,0) = \langle -|A|  \psi(t) \rangle_x = \langle -|A e ^{-tH}| x\rangle
-\end{equation}
-\item
-\underline{Since $[H,K^0] = 0$} there exist a basis to study the stochastic process associated to
-$H$ where \underline{$K^0$ is diagonal}. We might consider the following representation
-\begin{eqnarray}
-\label{Koper2}
-K^+_i|\xi\rangle &=& \left(\frac{1}{2} + \xi\right)  |\xi+1\rangle\nonumber \\
-K^-_i|\xi\rangle &=& \xi  |\xi-1\rangle\nonumber \\
-K^o_i|\xi\rangle &=& \left(\xi + \frac{1}{4}\right)  |\xi\rangle
-\end{eqnarray}
-where $|\xi\rangle$ denotes a vector which together with its transposed
-$\langle \xi|$ form a complete basis of a Hilbert space and its dual, that is
-\begin{equation}
-\langle \xi|\eta \rangle = \delta_{\xi,\eta}
-\end{equation}
-If we use this representation then
-$$
-H = -{\cal L}^*
-$$
-and the probability mass function for the $\Xi(t)$ process is encoded in
-the state $|\phi(t)\rangle$, namely
-\begin{equation}
-|\phi(t) \rangle = \sum_{\xi} P(\xi,t) |\xi\rangle
-\end{equation}
-It immediately follows that
-\begin{equation}
-\langle \xi|\phi(t) \rangle = P(\xi,t)
-\end{equation}
-To compute expectation with respect to the $\Xi(t)$ process
-we introduce the flat state
-\begin{equation}
-\langle -_{dual} | = \sum_{\xi} \;\langle \xi|
-\end{equation}
-which is such that
-\begin{equation}
-\langle -_{dual} | \xi\rangle = 1
-\end{equation}
-Then for any observable $A=A(\Xi(t))$ we have that its expectation value
-at time $t$ can be written as
-\begin{equation}
-\langle A(t) \rangle_\xi = \sum_{\eta}\,A(\eta)\, p(\eta,t;\xi,0) = \langle -_{dual}|A|  \phi(t) \rangle_{\xi} = \langle -_{dual}|A e ^{-tH}| \xi\rangle
-\end{equation}
-\item
-\underline{The claim is the following: Duality, in general, is going from the basis
-where}\\
-\underline{one generator of the group is diagonal to a basis where another generator of}\\
-\underline{ the group is diagonal.}
-
-In our case we change from a basis where $K^+$ is diagonal to the base where $K^0$ is diagonal.
-
-\begin{eqnarray}
-\langle - |\prod_i\frac{(2K_i^+)^{\xi_i}}{(2\xi_i-1)!!}|\psi(t)\rangle_x
-& = &
-\int dy \; \langle y |\prod_i\frac{(2K_i^+)^{\xi_i}}{(2\xi_i-1)!!} e^{tL^*}|x\rangle \nonumber \\
-& = &
-\sum_{\eta} \int dy \; \langle y |\prod_i\frac{(2K_i^+)^{\eta_i}}{(2\eta_i-1)!!}e^{tL^*}|x\rangle \langle \eta|\xi\rangle\nonumber \\
-& = &
-\sum_{\eta} \int dy \; \langle y| \otimes \langle \eta| \prod_i\frac{(2K_i^+)^{\eta_i}}{(2\eta_i-1)!!} e^{tL^*} | x\rangle \otimes|\xi\rangle\nonumber \\
-& = &
-\sum_{\eta} \int dy \; \langle x| \otimes \langle \xi | e^{tL} \prod_i\frac{(2K_i^+)^{\eta_i}}{(2\eta_i-1)!!} | y\rangle \otimes|\eta\rangle \nonumber \\
-& = &
-\sum_{\eta} \int dy \; \langle x| \otimes \langle \xi | e^{tL} \prod_i\frac{y^{2\eta_i}}{(2\eta_i-1)!!} | y\rangle \otimes|\eta \rangle \nonumber \\
-& = &
-\sum_{\eta} \int dy \; \langle x| \otimes \langle \xi | e^{t{\cal L}} \prod_i\frac{y_i^{2\eta_i}}{(2\eta_i-1)!!} | y\rangle \otimes|\eta \rangle \nonumber \\
-& = &
-\sum_{\eta} \int dy \; \langle x| \otimes \langle \xi | e^{t{\cal L}} \prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!} | y\rangle \otimes|\eta \rangle \nonumber \\
-& = &
-\sum_{\eta} \int dy \; \langle y| \otimes \langle \eta |\prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!} e^{t{\cal L}^*}  | x\rangle \otimes|\xi \rangle\nonumber \\
-& = &
-\sum_{\eta} \int dy \; \langle \eta | \prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!} e^{t{\cal L}^*}  |\xi \rangle   \langle y | x\rangle   \nonumber \\
-& = &
-\sum_{\eta} \langle \eta | \prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!} |\phi(t)\rangle_{\xi}  \nonumber \\
-& = &
-\langle -_{dual} |\prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!}|\phi(t)\rangle_{\xi}
-\end{eqnarray}
-
-
-\end{itemize}
-
-\section{General k}
-
-A convenient $(2j+1)$-dimensional representation of the SU(2) algebra is given by
-\begin{eqnarray}
-J^+_i |n_i\rangle &=& (2j-n_i) |n_i+1\rangle \nonumber \\
-J^-_i |n_i\rangle &=& n_i      |n_i-1\rangle \nonumber \\
-J^0_i |n_i\rangle &=& (n_i-j)  |n_i\rangle
-\end{eqnarray}
-where the quantum numbers $n_i\in\{0,1,\ldots,2j\}$.
-{\bf Note that in this representation the adjoint of $J^+_i$ is not
-$J^-_i$, UNLESS $j=1/2$}.
-
-A matrix representation is:
-$$
-J^+ = \left(
-\begin{array}{cccc}
- 0  &        &        &   \\
- 2j & \ddots &        &   \\
-    & \ddots & \ddots &   \\
-    &        &    1   &  0\\
-\end{array}\right)
-\qquad
-J^- = \left(
-\begin{array}{cccc}
- 0  &  1     &        &     \\
-    & \ddots & \ddots &     \\
-    &        & \ddots &  2j \\
-    &        &        &  0  \\
-\end{array}\right)
-\qquad
-J^0 = \left(
-\begin{array}{cccc}
- -j  &        &        &   \\
-     & \ddots &        &   \\
-     &        & \ddots &   \\
-     &        &        &  j\\
-\end{array}\right)
-$$
-
-In the SU(1,1) case one can use the infinite dimensional representation
-\begin{eqnarray}
-\label{newrepresentationsu11}
-K^+_i |n_i\rangle &=& (2k+n_i) |n_i+1\rangle \nonumber \\
-K^-_i |n_i\rangle &=& n_i      |n_i-1\rangle \nonumber \\
-K^0_i |n_i\rangle &=& (n_i+k)  |n_i\rangle
-\end{eqnarray}
-where the quantum numbers $n_i\in\{0,1,2,\ldots\}$.
-A matrix representation is:
-$$
-K^+ = \left(
-\begin{array}{cccc}
- 0  &        &        &   \\
- 2k & \ddots &        &   \\
-    & 2k+1   & \ddots &   \\
-    &        & \ddots & \ddots\\
-\end{array}\right)
-\qquad
-K^- = \left(
-\begin{array}{cccc}
- 0  &  1     &        &     \\
-    & \ddots &   2    &     \\
-    &        & \ddots &  \ddots \\
-    &        &        &  \ddots \\
-\end{array}\right)
-\qquad
-K^0 = \left(
-\begin{array}{cccc}
- k   &        &        &   \\
-     &  k+1   &        &   \\
-     &        & k+2    &   \\
-     &        &        &  \ddots\\
-\end{array}\right)
-$$
-Let's check that in this representation the operator is stochastic.
-I will do it for the bulk:
-\begin{eqnarray}
-L_{i,i+1}|n_i,n_{i+1}\rangle
-&=&
-(2k+n_i) n_{i+1}|n_i +1 ,n_{i+1}-1\rangle \nonumber\\
-&+&
-n_i(2k+n_{i+1})|n_i -1 ,n_{i+1}+1\rangle \nonumber\\
-&+&
-(-2(n_i+k)(n_{i+1}+k)+2k^2)|n_i,n_{i+1}\rangle
-\end{eqnarray}
-The sum of the rates is
-$$
-(2k+n_i) n_{i+1}+
-n_i(2k+n_{i+1})
--2(n_i+k)(n_{i+1}+k)+2k^2 =0
-$$
-
-
-
-
-
-
-
-
-
-% \end{document}
-- 
cgit v1.3


From 274943e723eb96f8c53a4abf5e090befa66f9cca Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 30 Dec 2020 14:01:29 +0100
Subject: Switched to biblatex.

---
 cover.tex | 46 ++++++++++++++++------------------------------
 1 file changed, 16 insertions(+), 30 deletions(-)

(limited to 'cover.tex')

diff --git a/cover.tex b/cover.tex
index 2d2fc10..ec9ff7a 100644
--- a/cover.tex
+++ b/cover.tex
@@ -1,25 +1,5 @@
 \documentclass[a4paper]{letter}
 
-\makeatletter
-\newenvironment{thebibliography}[1]
-     {\list{\@biblabel{\@arabic\c@enumiv}}%
-           {\settowidth\labelwidth{\@biblabel{#1}}%
-            \leftmargin\labelwidth
-            \advance\leftmargin\labelsep
-            \usecounter{enumiv}%
-            \let\p@enumiv\@empty
-            \renewcommand\theenumiv{\@arabic\c@enumiv}}%
-      \sloppy
-      \clubpenalty4000
-      \@clubpenalty \clubpenalty
-      \widowpenalty4000%
-      \sfcode`\.\@m}
-     {\def\@noitemerr
-       {\@latex@warning{Empty `thebibliography' environment}}%
-      \endlist}
-\newcommand\newblock{\hskip .11em\@plus.33em\@minus.07em}
-\makeatother
-
 \usepackage[utf8]{inputenc} % why not type "Bézout" with unicode?
 \usepackage[T1]{fontenc} % vector fonts plz
 \usepackage{newtxtext,newtxmath} % Times for PR
@@ -31,8 +11,12 @@
   linkcolor=purple
 ]{hyperref} % ref and cite links with pretty colors
 \usepackage{xcolor}
+\usepackage[style=phys]{biblatex}
+
+\addbibresource{bezout.bib}
 
 \signature{
+  \vspace{-3.5em}
   Jaron Kent-Dobias \& Jorge Kurchan
 }
 
@@ -53,16 +37,16 @@
 
 \opening{}
 
-The subject of `Complex Landscapes,' which started in the spin-glass
+The subject of `complex landscapes,' which started in the spin-glass
 literature, is concerned with functions (landscapes) of many variables, having
-a multiplicity of minimums, which are the objects of interest. Apart from its
-obvious interest for glassy systems, it has found a myriad applications  in
-many domains:  Computer Science, Ecology, Economics, Biology
+a multiplicity of minima, which are the objects of interest. Apart from its
+obvious interest for glassy systems, it has found a myriad applications in many
+domains: computer science, ecology, economics, biology
 \cite{Mezard_2009_Information}.
 
 In the last few years, a renewed interest has developed for landscapes for
 which the variables are complex. There are a few reasons for this: {\em i)} in
-Computational Physics, there is the main obstacle of the `sign problem', and a
+computational physics, there is the main obstacle of the `sign problem', and a
 strategy has emerged to attack it deforming the sampling space into complex
 variables. This is a most natural and promising path, and any progress made
 will have game-changing impact in solid state physics and lattice-QCD
@@ -73,17 +57,19 @@ concerning the very definition of quantum mechanics, which requires also that
 one move into the complex plane. 
 
 In all these cases, just like in the real case, one needs to know the structure
-of the `landscape', where are the saddle points and how they are connected,
-typical questions of `complexity'.  However, to the best of our knowledge,
+of the `landscape.' where are the saddle points and how they are connected,
+typical questions of `complexity.'  However, to the best of our knowledge,
 there are no studies extending the methods of the theory of complexity to
 complex variables.  We believe our paper will open a field that may find
 numerous applications and will widen our theoretical view of complexity in
 general.
 
 \closing{Sincerely,}
-\end{letter}
 
-\bibliographystyle{unsrt}
-\bibliography{bezout}
+\vspace{1em}
+
+\printbibliography[heading=none]
+
+\end{letter}
 
 \end{document}
-- 
cgit v1.3


From 9f0171b52574d8e63fc9178d36b41e78346e5805 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 30 Dec 2020 14:09:54 +0100
Subject: Fancy footnote citations.

---
 cover.tex | 14 ++++++--------
 1 file changed, 6 insertions(+), 8 deletions(-)

(limited to 'cover.tex')

diff --git a/cover.tex b/cover.tex
index ec9ff7a..a20ec7f 100644
--- a/cover.tex
+++ b/cover.tex
@@ -41,18 +41,18 @@ The subject of `complex landscapes,' which started in the spin-glass
 literature, is concerned with functions (landscapes) of many variables, having
 a multiplicity of minima, which are the objects of interest. Apart from its
 obvious interest for glassy systems, it has found a myriad applications in many
-domains: computer science, ecology, economics, biology
-\cite{Mezard_2009_Information}.
+domains: computer science, ecology, economics, biology.
+\footfullcite{Mezard_2009_Information}
 
 In the last few years, a renewed interest has developed for landscapes for
 which the variables are complex. There are a few reasons for this: {\em i)} in
 computational physics, there is the main obstacle of the `sign problem', and a
 strategy has emerged to attack it deforming the sampling space into complex
 variables. This is a most natural and promising path, and any progress made
-will have game-changing impact in solid state physics and lattice-QCD
-\cite{Cristoforetti_2012_New,Scorzato_2016_The}.  {\em ii)} At a more basic
-level, following the seminal work of E. Witten
-\cite{Witten_2010_A,Witten_2011_Analytic}, there has been a flurry of activity
+will have game-changing impact in solid state physics and lattice-QCD.
+\footfullcite{Cristoforetti_2012_New, Scorzato_2016_The}  {\em ii)} At a more basic
+level, following the seminal work of E.~Witten,
+\footfullcite{Witten_2010_A,Witten_2011_Analytic} there has been a flurry of activity
 concerning the very definition of quantum mechanics, which requires also that
 one move into the complex plane. 
 
@@ -68,8 +68,6 @@ general.
 
 \vspace{1em}
 
-\printbibliography[heading=none]
-
 \end{letter}
 
 \end{document}
-- 
cgit v1.3


From 5352b6f5885f1d46dfd16d8e998e9ed4a42f76c1 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 30 Dec 2020 14:13:03 +0100
Subject: Added salutation.

---
 cover.tex | 7 ++++---
 1 file changed, 4 insertions(+), 3 deletions(-)

(limited to 'cover.tex')

diff --git a/cover.tex b/cover.tex
index a20ec7f..3c90986 100644
--- a/cover.tex
+++ b/cover.tex
@@ -16,14 +16,15 @@
 \addbibresource{bezout.bib}
 
 \signature{
-  \vspace{-3.5em}
+  \vspace{-6\medskipamount}
+  \smallskip
   Jaron Kent-Dobias \& Jorge Kurchan
 }
 
 \address{
   Laboratoire de Physique \\
   Ecole Normale Sup\'erieure \\
-  24, rue Lhomond \\ 
+  24 rue Lhomond \\ 
   75005 Paris
 }
 
@@ -35,7 +36,7 @@
   Ridge, NY 11961
 }
 
-\opening{}
+\opening{To the editors of Physical Review,}
 
 The subject of `complex landscapes,' which started in the spin-glass
 literature, is concerned with functions (landscapes) of many variables, having
-- 
cgit v1.3


From 8373f749bbc33d63bfa2993cdba0c3dceb74fe50 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 30 Dec 2020 14:15:51 +0100
Subject: Small bib fixes.

---
 bezout.bib | 4 ++--
 cover.tex  | 2 +-
 2 files changed, 3 insertions(+), 3 deletions(-)

(limited to 'cover.tex')

diff --git a/bezout.bib b/bezout.bib
index 9e600f0..ba871fb 100644
--- a/bezout.bib
+++ b/bezout.bib
@@ -314,7 +314,7 @@
  volume = {251},
  url = {https://doi.org/10.22323%2F1.251.0016},
  doi = {10.22323/1.251.0016},
- booktitle = {Proceedings of The 33rd International Symposium on Lattice Field Theory (LATTICE 2015)},
+ booktitle = {Proceedings of The 33rd International Symposium on Lattice Field Theory ({LATTICE} 2015)},
  series = {Proceedings of Science}
 }
 
@@ -369,7 +369,7 @@
  pages = {347--446},
  url = {https://doi.org/10.1090%2Famsip%2F050%2F19},
  doi = {10.1090/amsip/050/19},
- booktitle = {Chern-Simons Gauge Theory: 20 Years After},
+ booktitle = {{Chern}-{Simons} Gauge Theory: 20 Years After},
  editor = {Andersen, Jørgen E. and Boden, Hans U. and Hahn, Atle and Himpel, Benjamin},
  series = {AMS/IP Studies in Advanced Mathematics}
 }
diff --git a/cover.tex b/cover.tex
index 3c90986..c721c52 100644
--- a/cover.tex
+++ b/cover.tex
@@ -53,7 +53,7 @@ variables. This is a most natural and promising path, and any progress made
 will have game-changing impact in solid state physics and lattice-QCD.
 \footfullcite{Cristoforetti_2012_New, Scorzato_2016_The}  {\em ii)} At a more basic
 level, following the seminal work of E.~Witten,
-\footfullcite{Witten_2010_A,Witten_2011_Analytic} there has been a flurry of activity
+\footfullcite{Witten_2010_A, Witten_2011_Analytic} there has been a flurry of activity
 concerning the very definition of quantum mechanics, which requires also that
 one move into the complex plane. 
 
-- 
cgit v1.3


From cf3f83827d960a70383232b65bef72398646fc10 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 30 Dec 2020 14:20:57 +0100
Subject: Made citations links invisible.

---
 cover.tex | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

(limited to 'cover.tex')

diff --git a/cover.tex b/cover.tex
index c721c52..b414930 100644
--- a/cover.tex
+++ b/cover.tex
@@ -6,9 +6,9 @@
 \usepackage[
   colorlinks=true,
   urlcolor=purple,
-  citecolor=purple,
-  filecolor=purple,
-  linkcolor=purple
+  linkcolor=black,
+  citecolor=black,
+  filecolor=black
 ]{hyperref} % ref and cite links with pretty colors
 \usepackage{xcolor}
 \usepackage[style=phys]{biblatex}
-- 
cgit v1.3


From 50767662937073c6804888dd25ab49420c7a939e Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 30 Dec 2020 14:30:29 +0100
Subject: Wording edits.

---
 cover.tex | 37 ++++++++++++++++++-------------------
 1 file changed, 18 insertions(+), 19 deletions(-)

(limited to 'cover.tex')

diff --git a/cover.tex b/cover.tex
index b414930..6e21d10 100644
--- a/cover.tex
+++ b/cover.tex
@@ -39,27 +39,26 @@
 \opening{To the editors of Physical Review,}
 
 The subject of `complex landscapes,' which started in the spin-glass
-literature, is concerned with functions (landscapes) of many variables, having
-a multiplicity of minima, which are the objects of interest. Apart from its
-obvious interest for glassy systems, it has found a myriad applications in many
-domains: computer science, ecology, economics, biology.
-\footfullcite{Mezard_2009_Information}
+literature, is concerned with functions (landscapes) of many variables having
+a multiplicity of minima. Apart from its obvious relevance to glassy systems,
+it has found applications in many domains: computer science, ecology,
+economics, and biology, to name a few.  \footfullcite{Mezard_2009_Information}
 
-In the last few years, a renewed interest has developed for landscapes for
-which the variables are complex. There are a few reasons for this: {\em i)} in
-computational physics, there is the main obstacle of the `sign problem', and a
-strategy has emerged to attack it deforming the sampling space into complex
-variables. This is a most natural and promising path, and any progress made
-will have game-changing impact in solid state physics and lattice-QCD.
-\footfullcite{Cristoforetti_2012_New, Scorzato_2016_The}  {\em ii)} At a more basic
-level, following the seminal work of E.~Witten,
-\footfullcite{Witten_2010_A, Witten_2011_Analytic} there has been a flurry of activity
-concerning the very definition of quantum mechanics, which requires also that
-one move into the complex plane. 
+In the last few years, a renewed interest has developed in landscapes for
+which the variables are complex. There are a couple reasons for this: in
+computational physics the `sign problem' is a major obstacle, and a strategy
+has emerged to attack it by deforming the sampling space into complex variables.
+This is a most natural and promising path, and any progress made will have
+game-changing impact in solid state physics and lattice QCD.
+\footfullcite{Cristoforetti_2012_New, Scorzato_2016_The} At a more basic
+level, following the seminal work of E.~Witten, \footfullcite{Witten_2010_A,
+Witten_2011_Analytic} there has been a flurry of activity concerning the very
+definition of quantum mechanics, which also requires that one move into the
+complex plane. 
 
-In all these cases, just like in the real case, one needs to know the structure
-of the `landscape.' where are the saddle points and how they are connected,
-typical questions of `complexity.'  However, to the best of our knowledge,
+In these cases, just as in the real case, one needs to understand the structure
+of the `landscape,' like the location of saddle points, how they are connected,
+and typical questions of `complexity.'  However, to the best of our knowledge,
 there are no studies extending the methods of the theory of complexity to
 complex variables.  We believe our paper will open a field that may find
 numerous applications and will widen our theoretical view of complexity in
-- 
cgit v1.3


From 569cd75da9eb1de4adbd8d2294cb9b3686923a82 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 30 Dec 2020 14:33:10 +0100
Subject: More wording edits.

---
 cover.tex | 21 ++++++++++-----------
 1 file changed, 10 insertions(+), 11 deletions(-)

(limited to 'cover.tex')

diff --git a/cover.tex b/cover.tex
index 6e21d10..6f74b46 100644
--- a/cover.tex
+++ b/cover.tex
@@ -44,17 +44,16 @@ a multiplicity of minima. Apart from its obvious relevance to glassy systems,
 it has found applications in many domains: computer science, ecology,
 economics, and biology, to name a few.  \footfullcite{Mezard_2009_Information}
 
-In the last few years, a renewed interest has developed in landscapes for
-which the variables are complex. There are a couple reasons for this: in
-computational physics the `sign problem' is a major obstacle, and a strategy
-has emerged to attack it by deforming the sampling space into complex variables.
-This is a most natural and promising path, and any progress made will have
-game-changing impact in solid state physics and lattice QCD.
-\footfullcite{Cristoforetti_2012_New, Scorzato_2016_The} At a more basic
-level, following the seminal work of E.~Witten, \footfullcite{Witten_2010_A,
-Witten_2011_Analytic} there has been a flurry of activity concerning the very
-definition of quantum mechanics, which also requires that one move into the
-complex plane. 
+Recently, interest has developed in landscapes for which the variables are
+complex. There are several reasons for this: in computational physics the
+`sign problem' is a major obstacle, and a strategy has emerged to attack it by
+deforming the sampling space into complex variables.  This is a most natural
+and promising path, and any progress made will have game-changing impact in
+solid state physics and lattice QCD.  \footfullcite{Cristoforetti_2012_New,
+Scorzato_2016_The} At a more basic level, following the seminal work of
+E.~Witten, \footfullcite{Witten_2010_A, Witten_2011_Analytic} there has been a
+flurry of activity concerning the very definition of quantum mechanics, which
+also requires that one move into the complex plane. 
 
 In these cases, just as in the real case, one needs to understand the structure
 of the `landscape,' like the location of saddle points, how they are connected,
-- 
cgit v1.3


From c3aef8d99961ccc06df11a4369bf112f483cc8cd Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 30 Dec 2020 15:13:52 +0100
Subject: Added name to return address.

---
 cover.tex | 7 ++++---
 1 file changed, 4 insertions(+), 3 deletions(-)

(limited to 'cover.tex')

diff --git a/cover.tex b/cover.tex
index 6f74b46..5abbdfe 100644
--- a/cover.tex
+++ b/cover.tex
@@ -22,9 +22,10 @@
 }
 
 \address{
-  Laboratoire de Physique \\
-  Ecole Normale Sup\'erieure \\
-  24 rue Lhomond \\ 
+  Jaron Kent-Dobias\\
+  Laboratoire de Physique\\
+  Ecole Normale Sup\'erieure\\
+  24 rue Lhomond\\ 
   75005 Paris
 }
 
-- 
cgit v1.3


From d13787cf9f35ab04eaf961f55cfbb8f1c75e423f Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 30 Dec 2020 15:43:29 +0100
Subject: Added sentence noted that the manuscript was modified.

---
 cover.tex | 3 +--
 1 file changed, 1 insertion(+), 2 deletions(-)

(limited to 'cover.tex')

diff --git a/cover.tex b/cover.tex
index 5abbdfe..4ebc19b 100644
--- a/cover.tex
+++ b/cover.tex
@@ -22,7 +22,6 @@
 }
 
 \address{
-  Jaron Kent-Dobias\\
   Laboratoire de Physique\\
   Ecole Normale Sup\'erieure\\
   24 rue Lhomond\\ 
@@ -62,7 +61,7 @@ and typical questions of `complexity.'  However, to the best of our knowledge,
 there are no studies extending the methods of the theory of complexity to
 complex variables.  We believe our paper will open a field that may find
 numerous applications and will widen our theoretical view of complexity in
-general.
+general. Our manuscript has been amended to emphasize these important connections with other areas of physics.
 
 \closing{Sincerely,}
 
-- 
cgit v1.3

