diff options
| -rw-r--r-- | frsb_kac-rice.tex | 13 |
1 files changed, 9 insertions, 4 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 3c6276d..6013f2f 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -119,7 +119,7 @@ when $\mu=\mu_m$, the critical points are marginal minima. =\mathcal D(\mu) +\operatorname*{extremum}_{\substack{R_d,D_d,\hat\epsilon\in\mathbb R\\\chi\in\Lambda}} \left\{ - \hat\epsilon\epsilon-\mu R_d + -\hat\epsilon\epsilon-\mu R_d +\frac12(2\hat\epsilon R_d-D_d)f'(1)+\frac12R_d^2f''(1) +\log R_d \right.\\\left. +\frac12\int_0^1dq\,\left( @@ -148,9 +148,14 @@ result is that, if the equilibrium state in the vicinity of zero temperature is given by a $k$-RSB ansatz, then the complexity is given by a $(k-1)$-RSB ansatz. Moreover, there is an exact correspondence between the parameters of the equilibrium saddle point in the limit of zero temperature and those of the -complexity saddle saddle at the ground state. If the equilibrium is given by $x_1,\ldots,x_k$ and $q_1,\ldots,q_k$, then the parameters $\tilde x_1,\ldots,\tilde x_{k-1}$ and $\tilde q_1,\ldots,\tilde q_{k-1}$ for the complexity in the ground state are +complexity saddle at the ground state. If the equilibrium is given by +$x_1,\ldots,x_k$ and $q_1,\ldots,q_k$, then the parameters $\tilde +x_1,\ldots,\tilde x_{k-1}$ and $\tilde q_1,\ldots,\tilde q_{k-1}$ for the +complexity in the ground state are \begin{align} - \tilde x_i=\frac1{\hat\epsilon}\lim_{\beta\to\infty}\beta x_i + \hat\epsilon=\lim_{\beta\to\infty}\beta x_k + && + \tilde x_i=\lim_{\beta\to\infty}\frac{x_i}{x_k} && \tilde q_i=\lim_{\beta\to\infty}q_i && @@ -366,7 +371,7 @@ The parameters: \begin{equation} S - =\mathcal D(\mu)+\hat\epsilon\epsilon+\lim_{n\to0}\frac1n\left( + =\mathcal D(\mu)-\hat\epsilon\epsilon+\lim_{n\to0}\frac1n\left( -\mu\sum_a^nR_{aa} +\frac12\sum_{ab}\left[ \hat\epsilon^2f(Q_{ab})+2\hat\epsilon R_{ab}f'(Q_{ab}) |