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| -rw-r--r-- | bezout.tex | 27 |
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@@ -53,10 +53,9 @@ defined by the energy: where the $J_{i_1\cdots i_p}$ are real Gaussian variables and the $z_i$ are real and constrained to a sphere $\sum_i z_i^2=N$. If there is a single term of a given $p$, this is known as the `pure $p$-spin' model, the case we shall -study here. Also in the Algebra \cite{Cartwright_2013_The} and Probability +study here. This problem has been studied also in the Algebra \cite{Cartwright_2013_The} and Probability literature \cite{Auffinger_2012_Random, Auffinger_2013_Complexity}. - -This problem has been attacked from several angles: the replica trick to +It has been attacked from several angles: the replica trick to compute the Boltzmann--Gibbs distribution \cite{Crisanti_1992_The}, a Kac--Rice \cite{Kac_1943_On, Rice_1939_The, Fyodorov_2004_Complexity} procedure (similar to the Fadeev--Popov integral) to compute the number of saddle-points of the @@ -66,13 +65,13 @@ high-energy configuration \cite{Cugliandolo_1993_Analytical}. Thanks to the relative simplicity of the energy, all these approaches are possible analytically in the large $N$ limit. -In this paper we shall extend the study to the case where $z\in\mathbb C^N$ -are and $J$ is a symmetric tensor whose elements are complex normal with +In this paper we shall extend the study to the case where the variables are complex +$z\in\mathbb C^N$ and $J$ is a symmetric tensor whose elements are 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 becomes $z^2=N$. The motivations for this paper are of two types. On the practical side, there -are situations in which complex variables have in a disorder problem appear +are indeed situations in which complex variables in a disorder problem appear naturally: such is the case in which they are {\em phases}, as in random laser problems \cite{Antenucci_2015_Complex}. Another problem where a Hamiltonian very close to ours has been proposed is the Quiver Hamiltonians @@ -85,9 +84,9 @@ underlying simplicity that is hidden in the purely real case. Consider, for example, the procedure of starting from a simple, known Hamiltonian $H_{00}$ and studying $\lambda H_{00} + (1-\lambda H_{0} )$, evolving adiabatically from $\lambda=0$ to $\lambda=1$, as is familiar from quantum annealing. The $H_{00}$ -is a polynomial of degree $p$ chosen to have simple, known roots. Because we -are working in complex variables, and the roots are simple all the way (we -shall confirm this), we may follow a root from $\lambda=0$ to $\lambda=1$. With +is a polynomial of degree $p$ chosen to have simple, known saddles. Because we +are working in complex variables, and the saddles are simple all the way (we +shall confirm this), we may follow a single one from $\lambda=0$ to $\lambda=1$, while with real variables minima of functions appear and disappear, and this procedure is not possible. The same idea may be implemented by performing diffusion in the $J$'s, and following the roots, in complete analogy with Dyson's stochastic @@ -97,12 +96,11 @@ This study also provides a complement to the work on the distribution of zeroes of random polynomials \cite{Bogomolny_1992_Distribution}. -Let us go back to our model. For the constraint we choose here $z^2=N$, + For our model the constraint we choose $z^2=N$, rather than $|z|^2=N$, in order to preserve the holomorphic nature of the functions. In addition, the nonholomorphic spherical constraint has a disturbing lack of critical points nearly everywhere, since $0=\partial^* -H=-p\epsilon z$ is only satisfied for $\epsilon=0$, as $z=0$ is forbidden by -the constraint. It is enforced using the method of Lagrange multipliers: +H=-p\epsilon z$ is only satisfied for $\epsilon=0$, as $z=0$ is forbidden by the constraint. It is enforced using the method of Lagrange multipliers: introducing the $\epsilon\in\mathbb C$, this gives \begin{equation} \label{eq:constrained.hamiltonian} H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right). @@ -110,6 +108,11 @@ introducing the $\epsilon\in\mathbb C$, this gives It is easy to see that {\em for a pure $p$-spin}, at any critical point $\epsilon=H/N$, the average energy. +Critical points are given by the set of equations: + +\begin{equation} +\frac{c_p}{(p-1)!}\sum_{ i, i_2\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p}, + Since $H$ is holomorphic, a point is a critical point of its real part if and only if it is also a critical point of its imaginary part. The number of |