By Michel Talagrand
This is a brand new, thoroughly revised, up to date and enlarged version of the author's Ergebnisse vol. forty six: "Spin Glasses: A problem for Mathematicians" in volumes (this is the second volume). within the eighties, a bunch of theoretical physicists brought a number of types for yes disordered platforms, referred to as "spin glasses". those versions are easy and quite canonical random constructions, of substantial curiosity for numerous branches of technological know-how (statistical physics, neural networks and laptop science). The physicists studied them via non-rigorous equipment and envisioned striking behaviors. This booklet introduces in a rigorous demeanour this intriguing new zone to the mathematically minded reader. It calls for no wisdom whatever of any physics. the current quantity II features a massive quantity of recent fabric, particularly all of the basic low-temperature effects got after the ebook of the 1st edition.
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Extra info for Mean Field Models for Spin Glasses: Volume II: Advanced Replica-Symmetry and Low Temperature
Example text
128) for u = i≤N ηi (σi − bi )/ N , so that u0 + u = √ i≤N ηi σi / N . 128) is immaterial. Before we move on, we need a simple lemma. 8. Consider a probability measure μ on R with density proportional to exp w(x) with respect to Lebesgue measure, where w is a concave function with w (x) ≤ −c, where c > 0. ) Assume that xdμ(x) = 0. v. 115). In words, we can witness a proportion of the variance of X by looking at ψ(X) instead of X. 7. 134) will be used to prove that ψ(σi − bi ) − ψ(σi − bi ) 2 ≥ 1 (σi − σi )2 .
4) where A is a monomial in the quantities u (Sv ), u (Sv ), u (Sv ) and where t 1 2 1 2 R1,2 = N −1 i 78) to the case τ < 0, α small. 3. (Level ≥ 2) Understand what happens for any τ < 0, α > 0. 4. (Level 2) If M/N ≥ α0 > α(τ ), for large N the set SN ∩ k≤M Uk is empty with probability at least 1 − exp(−N/K). 5. 1. 1, and we explain first the overall strategy. 1. 37) depends on κ, so that we denote it now by RSG(α, κ). 1 is that FN,M (κ) RSG(α, κ), hence 26 8. The Gardner Formula for the Sphere exp(−κ σ 2 )dσ exp(N RSG(α, κ)) . 80) CN,M Let us denote by Vol A the N -dimensional volume of a subset A of RN .