From 43ed1805b695086eb1eb7218fc483557a6df82be Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 12 Mar 2021 16:38:17 +0100 Subject: Changed some notation to be more clear. --- bezout.tex | 32 +++++++++++++------------------- 1 file changed, 13 insertions(+), 19 deletions(-) diff --git a/bezout.tex b/bezout.tex index eaa98ce..458b448 100644 --- a/bezout.tex +++ b/bezout.tex @@ -51,7 +51,7 @@ review see \cite{Castellani_2005_Spin-glass}) defined by the energy H_0 = \frac1{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p}, \end{equation} where $J$ is a symmetric tensor whose elements are real Gaussian variables and -$z\in\mathbb R^N$ is constrained to the sphere $z^2=N$. This problem has been +$z\in\mathbb R^N$ is constrained to the sphere $z^Tz=N$. This problem has been studied in the algebra \cite{Cartwright_2013_The} and probability literature \cite{Auffinger_2012_Random, Auffinger_2013_Complexity}. It has been attacked from several angles: the replica trick to compute the Boltzmann--Gibbs @@ -68,7 +68,7 @@ In this paper we extend the study to complex variables: we shall take $z\in\mathbb C^N$ and $J$ to be a symmetric tensor whose elements are \emph{complex} normal, with $\overline{|J|^2}=p!/2N^{p-1}$ and $\overline{J^2}=\kappa\overline{|J|^2}$ for complex parameter $|\kappa|<1$. The -constraint remains $z^2=N$. +constraint remains $z^Tz=N$. The motivations for this paper are of two types. On the practical side, there are indeed situations in which complex variables appear naturally in disordered @@ -98,27 +98,26 @@ of Lagrange multipliers: introducing $\epsilon\in\mathbb C$, our energy is \begin{equation} \label{eq:constrained.hamiltonian} H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right). \end{equation} -One might balk at the constraint $z^2=N$---which could appropriately be called -a \emph{hyperbolic} constraint---by comparison with $|z|^2=N$. The reasoning +One might balk at the constraint $z^Tz=N$---which could appropriately be called +a \emph{hyperbolic} constraint---by comparison with $z^\dagger z=N$. The reasoning behind the choice is twofold. First, we seek draw conclusions from our model that would be applicable to generic holomorphic functions without any symmetry. Samples of $H_0$ nearly provide this, save for a single anomaly: the value of the energy and its gradient at any point $z$ correlate along the $z$ direction, with -$\overline{H_0\partial_iH_0}\propto\kappa(z^2)^{p-1}z_i$ and -$\overline{H_0(\partial_iH_0)^*}\propto|z|^{2(p-1)}z_i$. Besides being a +$\overline{H_0\partial_iH_0}\propto \overline{H_0(\partial_iH_0)^*}\propto z_i$. Besides being a spurious correlation, in each sample there is also a `radial' gradient of magnitude proportional to the energy, since $z\cdot\partial H_0=pH_0$. This anomalous direction must be neglected if we are to draw conclusions about -generic functions, and the constraint surface $z^2=N$ is the unique surface +generic functions, and the constraint surface $z^Tz=N$ is the unique surface whose normal is parallel to $z$ and which contains the configuration space of the real $p$-spin model as a subspace. Second, taking the constraint to be the level set of a holomorphic function means the resulting configuration space is a \emph{bone fide} complex manifold, and therefore permits easy generalization of the integration techniques -referenced above. The same cannot be said for the space defined by $|z|^2=N$, +referenced above. The same cannot be said for the space defined by $z^\dagger z=N$, which is topologically the $(2N-1)$-sphere and cannot admit a complex structure. @@ -128,10 +127,10 @@ related problems have similar properties but are concerned with subspaces on which the energy is bounded. (In fact, identifying the appropriate subspace on which to define one's model often requires the study of critical points in the whole space.) Where it might be a problem, we introduce the additional -constraint $|z|^2\leq Nr^2$. The resulting configuration space is a complex +constraint $z^\dagger z\leq Nr^2$. The resulting configuration space is a complex manifold with boundary. We shall see that the `radius' $r$ proves an insightful knob in our present problem, revealing structure as it is varied. Note -that---combined with the constraint $z^2=N$---taking $r=1$ reduces the problem +that---combined with the constraint $z^Tz=N$---taking $r=1$ reduces the problem to that of the ordinary $p$-spin. The critical points are of $H$ given by the solutions to the set of equations @@ -211,8 +210,7 @@ be averaged independently. The $\delta$-functions are converted to exponentials by the introduction of auxiliary fields $\hat z=\hat x+i\hat y$. The average of those factors over $J$ can then be performed. A generalized Hubbard--Stratonovich allows a change of variables from the $4N$ original -and auxiliary fields to eight bilinears defined by $Na=|z|^2$, $N\hat a=|\hat -z|^2$, $N\hat c=\hat z^2$, $Nb=\hat z^*z$, and $Nd=\hat zz$ (and their +and auxiliary fields to eight bilinears defined by $Na=z^\dagger z$, $N\hat a=\hat z^\dagger\hat z$, $N\hat c=\hat z^T\hat z$, $Nb=\hat z^\dagger z$, and $Nd=\hat z^Tz$ (and their conjugates). The result, to leading order in $N$, is \begin{equation} \label{eq:saddle} \overline{\mathcal N}(\kappa,\epsilon,r) @@ -244,10 +242,7 @@ where $\theta=\frac12\arg\kappa$ and \begin{equation} C_{\pm}=\frac{a^p(1+p(a^2-1))\mp a^2|\kappa|}{a^{2p}\pm(p-1)a^p(a^2-1)|\kappa|-a^2|\kappa|^2}. \end{equation} -This leaves a single parameter, $a$, which dictates the magnitude of $|z|^2$, -or alternatively the magnitude $y^2$ of the imaginary part. The latter vanishes -as $a\to1$, where (as we shall see) one recovers known results for the real -$p$-spin. +This leaves a single parameter, $a$, which dictates the norm of $z$. The Hessian $\partial\partial H=\partial\partial H_0-p\epsilon I$ is equal to the unconstrained Hessian with a constant added to its diagonal. The eigenvalue @@ -385,9 +380,8 @@ in supersymmetric quiver theories \cite{Manschot_2012_From}. \includegraphics{fig/complexity.pdf} \caption{ The complexity of the 3-spin model at $\epsilon=0$ as a function of - the maximum `radius' $r=|z_{\mathrm{max}}|/\sqrt N$ at several values of - $\kappa$. The dashed line shows $\frac12\log(p-1)$, while the dotted shows - $\log(p-1)$. + the maximum `radius' $r$ at several values of $\kappa$. The dashed line + shows $\frac12\log(p-1)$, while the dotted shows $\log(p-1)$. } \label{fig:complexity} \end{figure} -- cgit v1.2.3-70-g09d2