Spin glasses and Stein’s method

Abstract

We introduce some applications of Stein's method in the high temperature analysis of spin glasses. Stein's method allows the direct analysis of the Gibbs measure without having to create a cavity. Another advantage is that it gives limit theorems with total variation error bounds, although the bounds can be suboptimal. A surprising byproduct of our analysis is a relatively transparent explanation of the Thouless-Anderson-Palmer system of equations. Along the way, we develop Stein's method for mixtures of two Gaussian densities.

Thouless–Anderson–Palmer equations for generic $p$-spin glasses

“…Indeed, once making this reduction, this is similar in spirit to the high temperature setting as in [7]. (This is stated and proved in a slightly more general setting in [7].) This step is shown in Section 2.…”

“…The first approach is to take · α as integration with respect to the Gibbs measure. This has been done by Talagrand [13] and Chatterjee [7] at sufficiently high temperature for the SK model where they establish (1.1) under this interpretation. A second approach, introduced by Bolthausen [6], is to interpret σ i α as a vector in high dimensions, and to understand (1.1) through a fixed point iteration scheme.…”

Quantitative stable limit theorems on the Wiener space

“…• Although there are many versions of Stein's method allowing one to deal with general continuous non-Gaussian targets (see, e.g., [3, 5-7, 12, 13, 32]), it seems that none of them can be reasonably applied to the limit theorems that are studied in this paper. Indeed, the above quoted contributions fall mainly in two categories: either those requiring that the density of the target distribution is explicitly known (and in this case the so-called "density approach" can be applied-see, e.g., [3,[5][6][7]), or those requiring that the target distribution is the invariant measure of some diffusion process (so that the "generator approach" can be used-see, e.g., [12,13,32]). In both instances, a detailed analytical description of the target distribution must be available.…”

“…In particular, in our framework no a priori knowledge of the distribution of S (and therefore of S · η) is required. One should note that in [3] one can find an application of Stein's method to the law of random objects with the form Sη, where η is a onedimensional Gaussian random variable and S has a law with a two-point support (of course, in this case the density of Sη can be directly computed by elementary arguments).…”

Central limit theorems for cavity and local fields of the Sherrington-Kirkpatrick model

“…Again, by applying Stein's method, Chatterjee [1] proved the first quantitative result regarding the limit law for the local fields that when k = 1 and U is a bounded measurable function U, the left-hand side of (13) has an error bound c(β 0 ) U ∞ / √ N, where c(β 0 ) is a constant depending on β 0 only. Thus, Theorem 3 improves this error bound if we require some smoothness on U.…”