From 487e9b69a43428aa1f8a19e0a16c228509e21786 Mon Sep 17 00:00:00 2001 From: "kurchan.jorge" Date: Fri, 11 Dec 2020 09:42:52 +0000 Subject: Update on Overleaf. --- bezout.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/bezout.tex b/bezout.tex index 1c416e2..fa973cc 100644 --- a/bezout.tex +++ b/bezout.tex @@ -336,7 +336,7 @@ values of $\kappa$ and $\epsilon$. Taking this saddle gives \log\overline{\mathcal N}(\kappa,\epsilon) =N\log(p-1). \end{equation} -This is precisely the Bézout bound, the maximum number of solutions that $N$ +This is, to this order, precisely the Bézout bound, the maximum number of solutions that $N$ equations of degree $p-1$ may have \cite{Bezout_1779_Theorie}. More insight is gained by looking at the count as a function of $a$, defined by $\overline{\mathcal N}(\kappa,\epsilon,a)=e^{Nf(a)}$. In the large-$N$ limit, this is the @@ -364,7 +364,7 @@ Notice that the limit of this expression as $a\to\infty$ corresponds with several values of $\kappa$ in Fig.~\ref{fig:complexity}. For any $|\kappa|<1$, the complexity goes to negative infinity as $a\to1$, i.e., as the spins are restricted to the reals. This is natural, given that the $y$ contribution to -the volume shrinks to zero as that of an $N$-dimensional sphere with volume +the volume shrinks to zero as that of an $N$-dimensional sphere $\sum_i y_i^2=N(a-1)$ with volume $\sim(a-1)^N$. However, when the result is analytically continued to $\kappa=1$ (which corresponds to real $J$) something novel occurs: the complexity has a finite value at $a=1$. Since the $a$-dependence gives a -- cgit v1.2.3-54-g00ecf