From a58f53debfe39ac7a098defef95e170cbcd360c0 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 9 Dec 2020 19:26:02 +0100 Subject: Added paragraph discussing the threshold. --- bezout.tex | 48 ++++++++++++++++++++++++++++++++++-------------- 1 file changed, 34 insertions(+), 14 deletions(-) (limited to 'bezout.tex') diff --git a/bezout.tex b/bezout.tex index 7fbe3cc..c586908 100644 --- a/bezout.tex +++ b/bezout.tex @@ -334,11 +334,13 @@ Notice that the limit of this expression as $a\to\infty$ corresponds with plotted as a function of $a$ for 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. -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 cumulative count, this -implies a $\delta$-function density of critical points along the line $y=0$. -The number of critical points contained within is +This is natural, given that the $y$ contribution to the volume shrinks to zero +as that of an $N$-dimensional sphere $\sim(a-1)^N$. 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 cumulative count, this implies a $\delta$-function +density of critical points along the line $y=0$. The number of critical points +contained within is \begin{equation} \lim_{a\to1}\lim_{\kappa\to1}\overline{\mathcal N}(\kappa,0,a) =(p-1)^{N/2}, @@ -358,15 +360,17 @@ limit as $\epsilon$ is varied. } \label{fig:complexity} \end{figure} -{\color{teal} {\bf somewhere else} - - -{\bf somewhere} In Figure \ref{fig:desert} we show that for $\kappa<1$ there is always a gap of $a$ close to one for which there are no solutions: this is natural, given that the $y$ contribution to the volume shrinks to zero as that of an $N$-dimensional sphere $\sim(a-1)^N$. -For the case $K=1$ -- i.e. the analytic continuation of the usual real computation -- the situation -is more interesting. In the range of values of $\Re H_0$ where there are exactly real solutions this gap closes, and this is only possible if the density of solutions diverges at $a=1$. -Another remarkable feature of the limit $\kappa=1$ is that there is still a gap without solutions around -`deep' real energies where there is no real solution. A moment's thought tells us that this is a necessity: otherwise a small perturbation of the $J$'s could produce a real, unusually deep solution for the real problem, in a region where we expect this not to happen. -} +These qualitative features carry over to nonzero $\epsilon$. In +Fig.~\ref{fig:desert} we show that for $\kappa<1$ there is always a gap of $a$ +close to one for which there are no solutions. For the case $\kappa=1$ -- the +analytic continuation to the usual real computation -- the situation is more +interesting. In the range of energies where there are real solutions this gap +closes, and this is only possible if the density of solutions diverges at +$a=1$. Another remarkable feature of the limit $\kappa=1$ is that there is +still a gap without solutions around `deep' real energies where there is no +real solution. A moment's thought tells us that this is a necessity: otherwise +a small perturbation of the $J$'s could produce a real, unusually deep solution +for the real problem, in a region where we expect this not to happen. \begin{figure}[htpb] \centering @@ -378,6 +382,22 @@ Another remarkable feature of the limit $\kappa=1$ is that there is still a gap } \label{fig:desert} \end{figure} +The relationship between the threshold and ground state energies is richer than +in the real case. In Fig.~\ref{fig:eggs} these are shown in the +complex-$\epsilon$ plane for several examples. Depending on the parameters, the +threshold always come at smaller magnitude than the ground state, or always +come at larger magnitude than the ground state, or change relative size as a +function of complex argument. For sufficiently large $a$ the threshold always +comes at larger magnitude than the ground state. If this were to happen in the +real case, it would likely imply our replica symmetric computation is unstable, +as having the ground state above the threshold would imply a ground state +Hessian with many negative eigenvalues, a contradiction with the notion of a +ground state. However, this is not a contradiction in the complex case, since +there cannot be minima of a complex function and the ground state therefore +takes on a different meaning. The relationship between the threshold, i.e., +where the gap appears, and the dynamics of, e.g., a minimization algorithm or +physical dynamics, are a problem we hope to address in future work. + \begin{figure}[htpb] \centering -- cgit v1.2.3-54-g00ecf