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?

1. 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 ${\overline {e^{\alpha X}}}=e^{\frac {\alpha ^{2}\sigma ^{2}}{2}}$.

2. Entropy contribution. What happens to the entropy of the model when the critical temperature is reached, and in the low temperature phase? What does this imply for the partition function $Z$?

3. Fluctuations, and back to average vs typical. Similarly to what we did for the entropy, one can define an annealed free energy $f^{A}$ from ${\overline {Z}}=e^{-N\beta f^{A}+o(N)}$: show that in the whole low-temperature phase this is smaller than the quenched free energy obtained above. Putting all the results together, justify why the average of the partition function in the low-T phase is "dominated by rare events".

Comment: the low-T phase of the REM is a frozen phase, characterized by the fact that the free energy is temperature independent, and that the typical value of the partition function is very different from the average value. In fact, the low-T phase is also $aglassphase$ in the sense discussed in the lecture. It is characterized by the fact that Replica Symmetry is broken, as one sees explicitly by re-deriving the free energy through the replica method. We go back to this in the next lectures/TDs.

Problem 3: the quenched free energy

1. Heavy tails and concentration. ccc

2. Inverse participation ratio. cccc