Research project description

Duration (months): 36 months

Main ERC field: PE - Physical Sciences and Engineering

ERC subfields: PE1_13 Probability, PE3_15 Statistical physics: phase transitions, noise and fluctuations, models of complex systems, etc. , PE1_16 Mathematical aspects of computer science

Key Words: GLASSY DYNAMICS AND METASTABILITY, RANDOM GRAPHS AND RANDOM MATRICES, RANDOM GEOMETRY AND RANDOM MEDIA, STOCHASTIC PARTICLE MODELS

Principal Investigator: MARTINELLI FABIO , Professore Ordinario , Università degli Studi ROMA TRE

Abstract

Large scale random structures consist of a large number of mutually interacting random components. Even if the interaction is local and simple, their collective behavior typically shows a host of intriguing phenomena like universality, anomalous fluctuations, phase transitions, complex time evolution. Their mathematical analysis requires the development of new powerful probabilistic ideas and techniques, combined with inputs from other branches of mathematics, notably analysis, geometric measure theory and combinatorics. In the last decade the field witnessed a stunning growth, with multi-year research programs (cf. NETWORKS in the Netherlands), outstanding ERC funding (cf. the ERC-AdG awarded to the PI in 2009-2012) and international yearly workshops (Oberwolfach).

We have selected some high impact problems where we expect to make a breakthrough. As it occurred already in the past, our findings will also impact other fields (e.g. statistical physics, theoretical computer science and economics). The team will consist of selected probabilists with a record of key achievements, a well established mutual collaboration and international recognition. Four main research lines have been chosen:

A: Glassy Dynamics and Metastability. The focus is on large scale stochastic dynamics, in particular interacting particle systems with constraints, bootstrap percolation and metastability. This is a very exciting and active field of research, with a wealth of applications across mathematics, physics, and biology.

B: Random Graphs and Random Matrices. The main themes are random walks on random graphs (mixing time,cutoff and invariant measure) and spectral analysis of random matrices associated to random graphs (universality and large deviations). The focus is on the, largely unexplored, non-reversible case corresponding to directed graphs.

C: Random Geometry and Random Media. This part of the project concerns the large scale behavior of specific random systems. Topics include basic disordered systems, random interfaces and random triangulations. We expect important outreach towards analysis, statistical physics and theoretical computer science.

D: Stochastic Particle Models. This section is articulated in two parts: the first, more analytical, features stochastic particles systems as random solutions to deterministic Burgers equations, scaling limit and KPZ universality; the second, more probabilistic, studies synchronization phenomena and stochastic games. Outreach towards biology and economics is expected.

Seminars and working groups will be run on a regular basis in order to impact on the scientific formation of young researchers (postdocs and graduate students). They will also be exposed to top international research environment in specific events (workshops and lectures) featuring outstanding international specialists. Special efforts will be made to have such key events across all the units.