CMSA Probability Seminar: A Gaussian convexity for logarithmic moment generating function

 Convex functions of Gaussian vectors are prominent objectives in many fields of mathematical studies. In this talk, I will establish a new convexity for the logarithmic moment generating function for this object and draw two consequences. The first leads to the Paouris-Valettas small deviation inequality that arises from the study of convex geometry. The second provides a quantitative bound for the Dotsenko-Franz-Mezard conjecture in the Sherrington-Kirkpatrick mean-field spin glass model, which states that the logarithmic anneal partition function of negative replica is asymptotically equal to the free energy.