Spin Glass to Paramagnetic Transition and Triple Point in Spherical SK Model

This paper studies spin glass to paramagnetic transition in the Spherical Sherrington-Kirkpatrick model with ferromagnetic Curie-Weiss interaction with coupling constant J and inverse temperature beta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}. The disorder of the system is represented by a general Wigner matrix. We confirm a conjecture of Baik and Lee (Stat Phys 165(2):185-224, 2016; Ann Henri Poincar & eacute; 18(6):1867-1917, 2017), that the critical window of temperatures for this transition is beta=1+bN-1/3logN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta = 1 + bN<^>{-1/3} \sqrt{\log N}$$\end{document} with b is an element of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b\in \mathbb {R}$$\end{document}. The limiting distribution of the scaled free energy is Gaussian for negative b and a weighted linear combination of independent Gaussian and Tracy-Widom components for positive b. In the special case where the Wigner matrix is from the Gaussian Orthogonal or Unitary Ensemble, we describe the triple point transition between spin glass, paramagnetic, and ferromagnetic regimes in a critical window for (beta,J)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\beta , J)$$\end{document} around the triple point (1, 1): the Tracy-Widom component is replaced by the one parameter family of deformations described by Bloemendal and Vir & aacute;g [9].