A Gaussian Convexity for Logarithmic Moment Generating Functions with Applications in Spin Glasses

2023/11/14 by Wei‐Kuo Chen, Chen, Wei-Kuo
Mathematics · Physics and Astronomy · #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2311.08351

Abstract

For any convex function F of n-dimensional Gaussian vector g with 𝔼 e^λF(g)<∞ for any λ>0, we show that λ^-1ln 𝔼 e^λF(g) is convex in λ∈ℝ. Based on this convexity, we draw three major consequences. The first recovers a version of the Paouris-Valettas lower deviation inequality for Gaussian convex functions with an improved exponent. The second establishes a quantitative bound for the Dotsenko-Franz-Mézard conjecture arising from the study of the Sherrington-Kirkpatrick (SK) mean-field spin glass model, which states that in the absence of external field, the annealed free energy of negative replica is asymptotically equal to the free energy. The last further establishes the differentiability for this annealed free energy with respect to the negative replica variable at any temperature and external field.

Related

• The convex structure of the Parisi formula for multi-species spin glasses
• Rigorous results on some simple spin glass models
• Uniqueness of Parisi measures for enriched convex vector spin glass
• Ordinary and logarithmical convexity of moment generating function
• Moderate Deviations for Gaussian Maxima and an Entropy Proof of Critical SK Free Energy Fluctuations