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diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index cbb33c3..5dff8b0 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -1055,12 +1055,25 @@ points. \subsection{A concrete example} -Consider two independent pure $p$ spin models $H_{p_1}({\mathbf s})$ and $H_{p_2}({\mathbf \sigma})$ of sizes $N$, and couple them weakly with $\varepsilon \; -{\mathbf \sigma} \cdot {\mathbf s}$. -The complexities are +One can construct a schematic 2RSB model from two 1RSB models. +Consider two independent pure $p$ spin models $H_{p_1}({\mathbf s})$ and +$H_{p_2}({\mathbf \sigma})$ of sizes $N$, and couple them weakly with +$\varepsilon \; +{\mathbf \sigma} \cdot {\mathbf s}$. The landscape of the pure models is much +simpler than that of the mixed because, in these models, fixing the stability +$\mu$ is equivalent to fixing the energy: $\mu=pE$. This implies that at each +energy level there is only one type of stationary point. Therefore, for the +pure models our formulas for the complexity and its Legendre transforms are +functions of one variable only, $E$, and each instance of $\mu^*$ inside mus be +replaced with $pE$. + +In the joint model, we wish to fix the total energy, not the energies of the +individual two models. Therefore, we insert a $\delta$-function containing +$(E_1+E_2)-E$ and integrate over $E_1$ and $E_2$. This results in a joint +complexity (and Legendre transform) \begin{eqnarray} - e^{N\Sigma(e)}&=&\int de_1 de_2 d\lambda \; e^{N[ \Sigma_1(e_1) + \Sigma_2(e_2) + O(\varepsilon) -\lambda N [(e_1+e_2)-e]}\nonumber \\ - e^{-G(\hat \beta)}&=&\int de de_1 de_2 d\lambda\; e^{N[-\hat \beta e+ \Sigma_1(e_1) + \Sigma_2(e_2) + O(\varepsilon) -\lambda N [(e_1+e_2)-e]} + e^{N\Sigma(E)}&=&\int dE_1\, dE_2\, d\lambda \, e^{N[ \Sigma_1(E_1) + \Sigma_2(E_2) + O(\varepsilon) -\lambda N [(E_1+E_2)-E]}\nonumber \\ + e^{NG(\hat \beta)}&=&\int dE\, dE_1\, dE_2\, d\lambda\, e^{N[-\hat \beta E+ \Sigma_1(E_1) + \Sigma_2(E_2) + O(\varepsilon) -\lambda N [(E_1+E_2)-E]} \end{eqnarray} The maximum is given by $\Sigma_1'=\Sigma_2'=\hat \beta$, provided it occurs in the phase in which both $\Sigma_1$ and $\Sigma_2$ are non-zero. The two systems are `thermalized', |