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 ${\vec {S}}=(S_{1},\cdots ,S_{N})$ satisfy the spherical constraint $\sum _{i=1}^{N}S_{i}^{2}=N$, and the energy associated to each configuration is $E({\vec {S}})=\sum _{1\leq i_{1}\leq i_{2}\leq \cdots i_{p}\leq N}J_{i_{1}\,i_{2}\cdots i_{p}}S_{i_{1}}S_{i_{2}}\cdots S_{i_{p}},$

  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 $p\geq 3$ is an integer.