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| -rw-r--r-- | frsb_kac-rice.tex | 15 |
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} |