Merge branch 'master' of https://git.overleaf.com/629a30c097d0b9f4b4f7a69d

1 files changed, 9 insertions, 6 deletions

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 5dff8b0..dda10f2 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -1095,13 +1095,15 @@ At $\hat \beta>\hat \beta_f$ there is a further transition.
 \subsection{\textit{R} and \textit{D}: response functions}
 
 The matrix field $R$ is related to responses of the stationary points to
-perturbations of the tensors $J$. Since the only dependence on $J$ lies in the
-measure, once the normalization $\mathcal N$ is replicated one finds
+perturbations of the tensors $J$. One adds to the Hamiltonian a random term $\varepsilon \tilde H_p = \varepsilon \sum_{i_1,...,i_p} \tilde J_{i_1,...,i_p} s_{i_1}...s_{i_p}$, where the $\tilde J$ are 
+random Gaussian uncorrelated with the $J$'s. 
+The response to these is:
 \begin{equation}
   \begin{aligned}
-    \frac1{N^p}\sum_{i_1\cdots i_p}\frac{\partial\langle s_{i_1}\cdots s_{i_p}\rangle}{\partial J^{(p)}_{i_1\cdots i_p}}
-    &=\lim_{n\to0}\frac1{N^p}\sum_{i_1\cdots i_p}\frac\partial{\partial J^{(p)}_{i_1\cdots i_p}}
-    \int\left(\prod_a^nd\nu(\mathbf s_a)\right)\,s^1_{i_1}\cdots s^1_{i_p} \\
+  & \overline{  \frac{\partial \langle \tilde H_p \rangle_{\tilde J}  }  {\partial \varepsilon} } 
+  %  \frac1{N^p}\sum_{i_1\cdots i_p}\frac{\partial\langle s_{i_1}\cdots s_{i_p}\rangle}{\partial J^{(p)}_{i_1\cdots i_p}}
+  %  &=\lim_{n\to0}\frac1{N^p}\sum_{i_1\cdots i_p}\frac\partial{\partial J^{(p)}_{i_1\cdots i_p}}
+  %  \int\left(\prod_a^nd\nu(\mathbf s_a)\right)\,s^1_{i_1}\cdots s^1_{i_p} \\
     & =\lim_{n\to0}\int\left(\prod_a^nd\nu(\mathbf s_a)\right)\sum_b^n\left[
       \hat\beta\left(\frac{\mathbf s_1\cdot\mathbf s_b}N\right)^p+
         p\left(-i\frac{\mathbf s_1\cdot\hat{\mathbf s}_b}N\right)\left(\frac{\mathbf s_1\cdot\mathbf s_b}N\right)^{p-1}
@@ -1111,7 +1113,8 @@ measure, once the normalization $\mathcal N$ is replicated one finds
 Taking the average of this expression over disorder and averaging over the equivalent replicas in the integral gives, similar to before,
 \begin{equation}
   \begin{aligned}
-    \overline{\frac1{N^p}\sum_{i_1\cdots i_p}\frac{\partial\langle s_{i_1}\cdots s_{i_p}\rangle}{\partial J^{(p)}_{i_1\cdots i_p}}}
+   \overline{  \frac{\partial \langle \tilde H_p \rangle_{\tilde J}  }  {\partial \varepsilon} } 
+  %  \overline{\frac1{N^p}\sum_{i_1\cdots i_p}\frac{\partial\langle s_{i_1}\cdots s_{i_p}\rangle}{\partial J^{(p)}_{i_1\cdots i_p}}}
     &=\lim_{n\to0}\int D[C,R,D]\,\frac1n\sum_{ab}^n(\hat\beta C_{ab}^p+pR_{ab}C_{ab}^{p-1})e^{nN\Sigma[C,R,D]}\\
     &=\hat\beta+pr_d-\int_0^1dx\,c^{p-1}(x)(\hat\beta c(x)+pr(x))
   \end{aligned}