Detailed description of the project
The project will be articulated around the main topics A, B, C and D described in the abstract. Each research line has its own targets (listed for convenience as A.1, A.2, etc), methodology and sought results. The high risk/high gain feature of the whole project, its possible outreach as well as its international relevance and its significance in terms of advancement of knowledge in fundamental research will emerge clearly.
A.We will mostly concentrate our efforts to push forward the mathematical analysis of Kinetically Constrained Models (KCM) and metastable dynamics in general. KCM pose extremely challenging mathematical problems because the constraints induce the existence of clusters of blocked sites, ergodicity breaking transition, multiple invariant measures, non-attractiveness and the failure of classic coercive inequalities. Moreover their numerical simulation is very delicate and produced in the past wrong conjectures and approaches as proved by the PI, A. Faggionato et al. There is therefore a strong demand of mathematical results in the physics community studying glasses. The main goals are as follows:
A.1 In '01 D. Aldous and P. Diaconis formulated a conjecture for the scaling limit of the stationary East model, a basic KCM in one dimension, as the vacancy density tends to zero. In different terms the conjecture appeared also in the physical literature [Evans et al. '13] motivated by Monte Carlo simulations. Implicit behind the conjecture there is a form of time scale separation for the relaxation times of the model. Time scale separation is now fully understood (Martinelli et al.[9]) and thanks to that we are confident to be able to resolve the conjecture, probably in the negative direction. Such a result would have a great impact also because it would show, once more, that numerical simulations of KCM's could be quite misleading.
A.2 In the East model on the integers a shape theorem holds: if at time t=0 the system has no vacancies then the set of vertices which have been updated within time t grows linearly in t [Blondel SPA '13], with normal fluctuations (Martinelli et al. [3]) around the front. In higher dimensions a shape theorem has been proved when the vacancy density is one [J. Martin, AoP '04]. In this case the East process becomes a last passage percolation problem. Proving an analogous result when the vacancy density is strictly below one is extremely challenging. In this context a first highly non trivial step is to prove that the speed of the front along any ray from the origin is larger than the speed along the coordinate axes. A partial result in this direction appeared already in [Martinelli et al. [2]].
A.3 It has been conjectured [J.P. Garrahan, J. Phys.: Cond. Matter. '02, M.J.Newman, Phys Rev. E '99] that the Glauber dynamics of certain ferromagnetic spin systems with multi-spin interaction, behave like a KCM at low temperature. However, the heuristic analysis suggesting this conjecture completely neglected key features of the dynamics and there is no consensus on the details of the analogy. Our aim is to put the above conjecture on firm mathematical grounds. We firstly plan to establish the correct Poincare' and logarithmic Sobolev inequalities by proving Dobrushin-Shlosman strong mixing conditions and secondly to nail down, via path techniques and capacity methods, the correct asymptotic of the relaxation times. Such a result would add important support to the assumption that KCM really capture several of the features of glassy dynamics.
A.4. A somewhat related metastable dynamics occurs also in the so called Inclusion Process (IP), an interacting particle system with mutual attraction between the particles. Because of the interaction the IP exhibits a condensation transition, with a positive fraction of all particles concentrated on a single site, randomly moving in space. The main goals are the following: in the reversible case we wish to describe the several relevant metastable time scales by means of potential theoretic techniques and to characterise the nucleation time using the martingale approach. In the non-reversible case we wish to analyse the condensation for a totally asymmetric dynamics on the torus. Key tools will come from Markov chain theory, general metastability theory, interacting particle systems, functional inequalities, capacity methods, large deviation theory, percolation and cellular automata. We will also sometimes use numerics as a preliminary tool to investigate new directions.
B.Natural models of random matrices are obtained by taking the adjacency matrix or the Laplacian of a random graph. The analysis of these random matrices can be used to provide insight in the relaxation to equilibrium of the associated random walks, and to explore specific features such as the cutoff phenomenon. These topics recently experienced a very rapid development in the context of undirected sparse random graphs, especially after the works of Lubeztky, Peres, Sly and others. In particular, these works established the cutoff phenomenon for regular graphs and for the giant component of a supercritical Erdos-Renyi graph. When the graph has directed edges, the associated random walks are non-reversible. In this case even a description of the invariant measure of the process can become rather challenging and the problem has essentially never been explored before. The main goals are as follows.
B.1 Convergence to equilibrium and cutoff phenomenon for a class of random directed graphs (digraphs). A first step was recently taken in this direction in the special case of sparse digraphs with given bounded degree sequences. Other models to be investigated include: (i) the Erdos-Renyi random digraph, (ii) the weighted directed graph with heavy-tailed random weights. As in the undirected case, we expect the analysis to be based on finding a suitable branching approximation for local neighborhoods. In the directed case however, the invariant measure being unknown, this seems to require a higher precision together with the development of refined martingale techniques. Given this state of affairs, even a small progress in any of the models mentioned above is likely to have a strong impact on the community.
B.2 Another related direction is concerned with the spectral analysis of the above mentioned random matrices. In this respect, we would like to make progress in the following directions: (i) convergence of the empirical spectral measure and (ii) proof of sharp estimates on the spectral radius. The first problem is related to an extension to the sparse case of the celebrated circular law for dense non-Hermitian matrices with i.i.d. entries. The second item aims at establishing a conjecture recently proposed by Bordenave, Caputo, Chafai indicating a sharp discrepancy in the behavior of the spectral radius between the Hermitian and the non-Hermitian setting. This line of research rests on more traditional random matrix theory techniques but the non-Hermitian character of the matrices associated to directed graphs makes it in many ways very challenging. As the recent spectacular developments around the circular law have shown, any tiny advancement in the non-Hermitian setting is predicted to have a very strong impact.
B.3 Large deviations for sparse random digraphs. This analysis was recently initiated by Caputo and Bordenave in the undirected case and their results have been used in the recent proof, after 25 years, of the satisfiability threshold for random k-SAT [J.Ding et al. '14]. We would like to explore the extension to the directed case. This requires in particular a detailed analysis of locally tree-like neighborhoods in directed random graphs, a topic that will certainly experience a rapid development in near future.
C.This line of research concerns the scaling limit of disordered systems and the large scale behavior of random surfaces and random triangulations. The main targets are:
C.1 As shown in Caravenna et al. [1], partition functions of "disorder relevant" random systems admit non-trivial scaling limits, in a continuum and weak disorder regime. This leads to the construction of "continuum disordered systems" which encodes universal properties for "directed models". We refer to (Caravenna et al. [2]) for pinning models and to [Alberts et al. JSP '14] for directed polymers. The analysis of "non-directed models" is more subtle and largely open. Our aim is to construct the "continuum 2d random field Ising model", extending to the disordered setting the work [Camia et al. AoP '15]. The groundbreaking results by [Hongler et al. Ann. Math. 2015] are expected to play a major role.
C.2 For "marginally relevant" system (characterized by logarithmic divergences) a common universal structure was recently revealed in [Caravenna et al., arXiv 2015], for both pinning models and directed polymers, in the "sub-critical regime". Our aim is to push the analysis to the very challenging "critical regime". This could be applied to the 2d stochastic heat equation, improving earlier results by [Bertini and Cancrini, J. Phys A '98]. Intriguing analogies with log-correlated random fields suggest that common techniques could be employed, alongside the more usual "polynomial chaos" expansions. Other relevant methodologies are expected to come from the theory of "noise sensitivity".
C.3 A natural class of random interfaces is characterized by a discrete height function on the vertex set of a box of the two dimensional lattice, together with a statistical weight for the heights configuration given by Exp[-ß ∑ |grad |^k.] Surfaces with discrete heights are quite hard to analyse and they behave very differently at low temperature (β>> 1) w.r.t. the continuous case . In (Martinelli et al. [7]) the exact asymptotics of the probability that a SOS surface (k=1) stays positive in a large box, an open question for many years, has been computed accurately using the scaling limit results of SOS [Martinelli et al. [1]]. We want to attack the same problem for the Discrete Gaussian model (k=2), for which, after more than twenty years, there has been recently important progress on concentration of its maximum and of the height above a floor [Martinelli et al. '15]. The main tools should be monotonicity, Peierls estimates, cluster expansion and local large deviations.
C.4 In the context of sharp interface limits, a somewhat related issue is the derivation of the motion by mean curvature from the (deterministic) Allen-Cahn equation. We consider the equation with a small random forcing term and we aim at finding its large deviations asymptotics in the sharp interface limit. The corresponding rate function should be the functional derived in [Kohn et. al. CPAM '07] in a purely variational setting. In particular, its zero level set consists of the motion by mean curvature in the Brakke's formulation.
C.5 Random lattice triangulations of boxes of the two dimensional lattice, with statistical weight λ^(length of triangulation), have been successfully analyzed in two cases: (i) λ<<1 and (ii) λ>1. The case of an arbitrary λ<1 has been recently attacked via Lyapunov function methods in [A. Stauffer, '15] and completely solved, for thin boxes, in [Martinelli et al. '15]. Still wide open is to provide a polynomial bound on the mixing time of the corresponding Glauber dynamics for any λ<1, our first target. Similarly, for λ>1 the goal is to get the correct exponential divergence of the mixing time, a problem related to finding the longest triangulation. Finally the third target is to investigate the Holy Grail for the subject, i.e. the uniform case λ=1, at least for thin rectangles.
D.As anticipated, this line of research has two faces: a more analytic one around the wide subject of KPZ-Burgers equations and KPZ universality class, and a more probabilistic one focused on synchronization phenomena and their connection with mean field games. The main goals are as follows.
D.1 Many features (e.g. correlation functions, scaling properties, invariant measures, marginal distributions) of the so-called "KPZ fixed point" have been investigated by exploiting explicit formulas derived for processes that are believed to belong to the KPZ universality class [Corwin et al. JSP '15]. At least at a formal level, it is possible to interpret the KPZ fixed point as a random, distribution-valued solution of the deterministic Burgers equation. Following the above approach, our main target is to better understand the KPZ fixed point by studying the long time behaviour of a new ad hoc interacting particle system, whose trajectories are in one-to-one correspondence with piecewise constant solutions of the deterministic Burgers equation. In this model particles are of two types, either repulsive or attractive, and are grouped in stacks. Stacks of particles perform a non-local deterministic dynamics, but they can also randomly split, merge and be created. As a first step we plan to study the long time evolution of the process (invariant measures, etc) and to define a numerical protocol for its simulation.
D.2 The first main target is to understand the emergence of rhythmic behavior in complex networks in some generality. Despite of the intense research on the subject, in particular for applications to neurosciences, a sufficiently wide picture indicating the origin of this phenomenon in some generality is still missing. Recent works indicate the role played by time delay ([J. Touboul 15], [S. Ditlevsen et al. '15]) and dissipation [Dai Pra et al. [1], [7]]; both may be interpreted as time-symmetry breaking of a reference stochastic dynamics. This point of view will be the subject of further investigation, including other possible origins of time-symmetry breaking, such as quenched randomness [L. Bertini et al., PTRF ’14] and forecasting. The latter is related to stochastic dynamic games and in particular to mean-field games, i.e. limit models for symmetric non-zero-sum non-cooperative N-player games with interaction of mean field type [O.Gueant et al. ’10]. This last topic, whose development has been formidable from the PDE point of view, still raises several open questions from the point of view of interacting particle systems, in particular for what concerns finite size effect [M. Fischer, ’15]. In this direction we plan to study large deviations and fluctuation theorems for the convergence to the limit of infinitely many players.