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| -rw-r--r-- | bezout.tex | 13 |
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@@ -23,12 +23,21 @@ \date\today \begin{abstract} - We study the saddle-points of the $p$-spin model, the best understood example of `complex (rugged) landscape' in the space in which - saturates the Bézout bound \cite{Bezout_1779_Theorie}. + We study the saddle-points of the $p$-spin model, the best understood example of `complex (rugged) landscape' in the space in which all its $N$ variables are allowed to be complex. The problem becomes + a system of $N$ random equations of degree $p-1$. + We solve for quantities averaged over randomness in the $N \rightarrow \infty$ limit. + We show that the number of solutions saturates the Bézout bound $\ln {\cal{N}}\sim N \ln (p-1) $\cite{Bezout_1779_Theorie}. +The Hessian of each saddle is given by a random matrix of the form $M=a A^2+b B^2 +ic [A,B]_-+ d [A,B]_+$, +where $A$ and $B$ are GOE matrices and $a-d$ real. Its spectrum has a transition +from one-cut to two-cut that generalizes the notion of `threshold level' that is well-known in the real problem. +In the case that the disorder is itself real, only the square-root of the total number solutions are real. +In terms of real and imaginary parts of the energy, the solutions are divided in sectors where the saddles have +different topological properties. \end{abstract} \maketitle + \begin{equation} \label{eq:bare.hamiltonian} 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} |