T-2

In this set of problems, we use the replica method to study the equilibrium properties of a prototypical toy model of glasses, the spherical $p$-spin model. In the spherical $p$-spin model the configurations ${\vec {\sigma }}=(\sigma _{1},\cdots ,\sigma _{N})$ that the system can take satisfy the spherical constraint $\sum _{i=1}^{N}\sigma _{i}^{2}=N$, and the energy associated to each configuration is

$E({\vec {\sigma }})=\sum _{1\leq i_{1}\leq i_{2}\leq \cdots i_{p}\leq N}J_{i_{1}\,i_{2}\cdots i_{p}}\sigma _{i_{1}}\sigma _{i_{2}}\cdots \sigma _{i_{p}},$

where the coupling constants $J_{i_{1}\,i_{2}\cdots i_{p}}$ are independent random variables with Gaussian distribution with zero mean and variance $p!/(2N^{p-1}),$ and $p\geq 3$ is an integer.

Problem 1: the annealed free energy

In TD1, we defined the quenched free energy density as the quantity controlling the scaling of the typical value of the partition function $Z$. The annealed free energy $f_{\rm {ann}}$ instead controls the scaling of the average value of $Z$. It is defined by

$f_{\rm {ann}}=-\lim _{N\to \infty }{\frac {1}{\beta N}}\log {\overline {Z}}.$

Let us compute this quantity.

1. Energy correlations. At variance with the REM, in the spherical $p$-spin the energies at different configurations are correlated. Show that ${\overline {E({\vec {\sigma }})E({\vec {\tau }})}}=Nq({\vec {\sigma }},{\vec {\tau }})^{p}/2+o(1)$, where $q({\vec {\sigma }},{\vec {\tau }})={\frac {1}{N}}\sum _{i=1}^{N}\sigma _{i}\tau _{i}$ is the overlap between the two configurations. Why an we say that for $p\to \infty$ this model converges with the REM discussed in the previous TD?

2. Energy contribution. Show that computing ${\overline {Z}}$ boils down to computing the average ${\overline {e^{-\beta J_{i_{1}\,\cdots i_{p}}\sigma _{i_{1}}\cdots \sigma _{i_{p}}}}}$. Compute this average. Hint: if X is a centered Gaussian variable with variance $\sigma ^{2}$, then ${\overline {e^{\alpha X}}}=e^{\frac {\alpha ^{2}\sigma ^{2}}{2}}$.

3. Entropy contribution. The volume of a sphere of radius ${\sqrt {N}}$ in dimension $N$ is given by $N^{\frac {N}{2}}\pi ^{\frac {N}{2}}/\left({\frac {N}{2}}\right)!$. Use the large-N asymptotic of this to conclude the calculation of the annealed free energy: the final result is only slightly different with respect to the free-energy of the REM in the high-temperature phase: can you identify the source of this difference?

Problem 2: the quenched free energy

The quenched free energy density is the quantity controlling the scaling of the typical value of the partition function $Z$. This means that

$f=-\lim _{N\to \infty }{\frac {1}{\beta N}}{\overline {\log Z}}.$

This formula differs from the one above by the order in which the logarithm and the average over disorder are taken.

1. Heavy tails and concentration. ccc

2. Inverse participation ratio. cccc