T-2

Goal: In this set of problems, we use the replica method to study the equilibrium properties of a prototypical mean-field model of glasses, the spherical $p$-spin model. We set up the calculation in these two problems, and conclude it next week.
Techniques: replica trick, Gaussian integrals.

Quenched vs annealed, and the replica trick

• Spherical p-spin model. In the spherical $p$-spin model the configurations $\overrightarrow{S}=(S_1,\cdots ,S_N)$ satisfy the spherical constraint $\sum _{i=1}^N{S}_i^2=N$, and the energy associated to each configuration is $E(\overrightarrow{S})=\sum _{1\le i_1\le i_2\le \cdots i_p\le N}{J}_{i_1{i}_2\cdots i_p}{S}_{i_1}{S}_{i_2}\cdots S_{i_p},$

  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 $p\ge 3$ is an integer.