ANR-DFG project REMECO

Project name: Random Energy Models: Extremes, Complex temperature, Optimization (REMECO)
Project numbers: ANR-20-CE92-0010-01; DFG Project-ID 446173099
Timeline: April 1, 2021 - March 31, 2024 (3 years)
Funding bodys: Deutsche Forschungsgemeinschaft (DFG), Agence Nationale de Recherche (ANR)
Host Institutions: Université de Toulouse III - Paul Sabatier, Institut de Mathématiques de Toulouse and Johannes Gutenberg-Universität Mainz, Institut für Mathematik
Coordinators: Lisa Hartung and Pascal Maillard (myself)

Project summary

This project aims at significantly advancing the understanding of the continuous random energy models (CREM), which are a class of toy models for strongly correlated random functions on a high-dimensional space such as mean-field spin glasses. We aim at investigating three different aspects: the distribution of its extreme values, the asymptotic behavior of the partition function at complex temperatures and the efficiency of optimization algorithms. The CREM can be described as a Gaussian process on a tree whose covariance is a function A of the overlap. We want to obtain a precise description of the extreme values (and their structure) in the case where A is strictly concave which is missing up to this point. Another goal is to use these insights to study the behavior of the partition function of the CREM at complex temperature (for strictly concave A) and not only show that the phase diagram has seven phases as has been conjectured but also obtain a precise description of the limit. Last but not least we want to study the efficiency of optimization algorithms. For instance, we wish to investigate the behavior close to the algorithmic hardness threshold the existence of which has been obtained recently.

Project-related publications (prior to start of the project)

• A. Bovier, L. Hartung, From 1 to 6: a finer analysis of perturbed branching Brownian motion, Communication in Pure and Applied Mathematics, 73(7), 1490-1525 (2020).
• A. Cortines, L. Hartung, O. Louidor, The structure of extreme level sets in branching Brownian motion, Ann. Probab. 47, No, 4 (2019), 2257-2302.
• L. Hartung and A. Klimovsky, The phase diagram of the complex branching Brownian motion energy model, Electron. J. Probab., Volume 23, paper no. 127, 27 pp. (2018)
• A. Bovier and L. Hartung, Variable speed branching Brownian motion 1. Extremal processes in the weak correlation regime, ALEA, Lat. Am. J. Probab. Math. Stat. 12, 261-291 (2015)
• A. Bovier, L. Hartung, The extremal process of two-speed branching Brownian motion, Electron. J. Probab. 19, No. 18, 1-28 (2014).
• L. Addario-Berry, P. Maillard, The algorithmic hardness threshold for continuous random energy models.Math. Stat. Learn., 2(1), 77–101 (2019).
• M. Pain, P. Maillard, 1-stable fluctuations in branching Brownian motion at critical temperature I: the derivative martingale. The Annals of Probability, 47(5), 2953-3002 (2019).
• P. Maillard, O. Zeitouni, Slowdown in branching Brownian motion with inhomogeneous variance. Ann. Inst. H. Poincaré Probab. Statist. 52, no. 3, 1144-1160 (2016).
• P. Maillard, Speed and fluctuations of N-particle branching Brownian motion with spatial selection. Probab. Th. Rel. Fields, 166, no. 3, 1061-1173 (2016).
• P. Maillard, A note on stable point processes occurring in branching Brownian motion. Electr. Comm. Probab., 18, no. 5, 1-9 (2013).