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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -spin model the configurations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z } . The annealed free energy $f_{\rm {ann}}$ instead controls the scaling of the average value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z } . It is defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{E(\vec{\sigma}) E(\vec{\tau})}= N q(\vec{\sigma}, \vec{\tau})^p/2 + o(1) } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{Z}} boils down to computing the average Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{e^{\alpha X}}=e^{\frac{\alpha^2 \sigma^2}{2} }} .

2. Entropy contribution. The volume of a sphere of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{N}} in dimension $N$ is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N^{\frac{N}{2}} \pi^{\frac{N}{2}}/(\frac{N}{2})!} . Use the large-N asymptotic of this to conclude the calculation of the annealed free energy.

Problem 3: the quenched free energy

1. Heavy tails and concentration. ccc

2. Inverse participation ratio. cccc