The extremal process of critical points of the pure p-spin spherical spin glass model

• 成果类型: Article
• 署名作者: Subag, Eliran; Zeitouni, Ofer
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-016-0724-2
• 发表日期: 2017
• 页码: 773-820
• 关键词: gaussian free-field; competing particle-systems; branching random-walk; brownian-motion; invariant-measures; minimal position; limit-theorem; maximum; energy; CONVERGENCE

摘要

Recently, sharp results concerning the critical points of the Hamiltonian of the p-spin spherical spin glass model have been obtained by means of moments computations. In particular, these moments computations allow for the evaluation of the leading term of the ground-state, i.e., of the global minimum. In this paper, we study the extremal point process of critical points-that is, the point process associated to all critical values in the vicinity of the ground-state. We show that the latter converges in distribution to a Poisson point process of exponential intensity. In particular, we identify the correct centering of the ground-state and prove the convergence in distribution of the centered minimum to a (minus) Gumbel variable. These results are identical to what one obtains for a sequence of i.i.d variables, correctly normalized; namely, we show that the model is in the universality class of REM.