The Random Energy Model

In an attempt to gain some understanding of the low temperature regime in mean-field spin glasses, Derrida [16] investigated the so-called Random Energy Model (see also [45]). Remark that in the SK-model (with h = 0, say), the Hamiltonian $$ H(\sigma ) = \frac{1} {{\sqrt N }}\sum\limits_{1 \leqslant i < j \leqslant N} {J_{i,j} \sigma _i \sigma _j } $$ is a family of random variables, indexed by the σ ∈ ΣN, which are normally distributed with variance (N — 1)/2. The main difficulty in the SK-model is hidden in the fact that these random variables are not independent. In fact, \(\mathbb{E}H\left( \sigma \right)H\left( {\sigma '} \right) = \left( {{{1} \left/ {2} \right.}N} \right)\left( {{{{\left\langle {\sigma ,\sigma '} \right\rangle }}^{2}} - N} \right)\). One may however ask whether some interesting features show up when assuming that these variables are just independent Gaussian random variables. There is no point to stick to the variance (N — 1)/2, and we take the variance N. Evidently then, the a need not carry any internal structure. We therefore assume that we have just 2N independent Gaussian random variables, say, \(X_{\alpha }^{{\left( N \right)}},1 \leqslant \alpha \leqslant {{2}^{N}}\) defined on some probability space, \(\left( {\Omega ,\mathcal{F},\mathbb{P}} \right)\) which are centered and have variance N. We define the “Gibbs measure” on the set ΣN of “configurations” α’s by defining for any ω ∈ Ω, and any β > 0 $$ P_{\omega ,\beta ,N} (\alpha ) = 2^{ - N} \frac{{\exp [\beta X_\alpha ^{(N)} \left( \omega \right)]}} {{Z_{\omega ,\beta ,N} }}, $$ where \({{Z}_{{\omega ,\beta ,N}}} = {{\Sigma }_{\alpha }}{{2}^{{ - N}}}\exp \left[ {\beta X_{\alpha }^{N}\left( \omega \right)} \right] = {{E}_{o}}{{e}^{{\beta X\cdot \left( \omega \right)}}}\), P o being again the uniform distribution on ΣN. For any fixed ω ∈ Ω, this is α probability distribution on the α, and we would like to know how it behaves for N → ∞, and almost all ω. The first task is to investigate the free energy $$ f\left( \beta \right) = \mathop {\lim }\limits_{N \to \infty } \frac{1} {N}\log z_{w,\beta ,N} . $$