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| -rw-r--r-- | frsb_kac_new.tex | 11 |
1 files changed, 8 insertions, 3 deletions
diff --git a/frsb_kac_new.tex b/frsb_kac_new.tex index 34ca3c9..514fda9 100644 --- a/frsb_kac_new.tex +++ b/frsb_kac_new.tex @@ -162,7 +162,7 @@ carefully treating the $k$th term in each sum separately from the rest, we get \right\} \end{aligned} \end{equation} -$F$ is a $k-1$ RSB ansatz with all eigenvalues scaled by $\tilde\beta z^{-1}$ and shifted by +This is a $k-1$ RSB ansatz with all eigenvalues scaled by $\tilde\beta z^{-1}$ and shifted by 1, with effective temperature $\tilde\beta$, and an extra term. This can be seen more clearly by rewriting the result in terms of the matrix $\tilde Q$, a $(k-1)$-RSB Parisi matrix built from the $\tilde x_1,\ldots,\tilde x_{k-1}$ and @@ -174,8 +174,13 @@ $q_1,\ldots,q_{k-1}$, which gives \right) \end{equation} {\em We have lost one level of RSB because at zero temperature the states become points.} - -{\bf Jaron: should'nt we put here the continuum solution?} +In the continuum case, this is +\begin{equation} \label{eq:ground.state.free.energy} + \lim_{\beta\to\infty}\tilde\beta F + =-\frac12z\tilde\beta f'(1)-\frac12\int dq\left( + \tilde\beta^2f''(q)\tilde\chi(q)+\frac1{\tilde\chi(q)+\tilde\beta z^{-1}} + \right) +\end{equation} |