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| -rw-r--r-- | bezout.tex | 26 |
1 files changed, 13 insertions, 13 deletions
@@ -24,18 +24,18 @@ \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 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 + example of a `complex' (rugged) landscape -- when its $N$ variables are + complex. These points are the solutions to a system of $N$ random equations + of degree $p-1$. We solve for $\overline{\mathcal{N}}$, the number of + solutions averaged over randomness in the $N\to\infty$ limit. We find that + it saturates the Bézout bound $\log\overline{\mathcal{N}}\sim N \log(p-1)$. + 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. The + results from the real problem are recovered in the limit of real disorder. In + this case, 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} @@ -53,7 +53,7 @@ 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. This problem has been studied 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}. It has been attacked from several angles: the replica trick to compute the Boltzmann--Gibbs distribution \cite{Crisanti_1992_The}, a Kac--Rice |