DISORDERED SYSTEMS, RANDOM SPATIAL PROCESSES AND SOME APPLICATIONS

I will introduce the Monomer-Dimer model in Statistical Mechanics, and show its solution on the Erdos-Rényi random graph (based on a joint work with P. Contucci). This result relies on the fact that such a graph locally looks like a Galton-Watson tree. Beyond this theorem of Probability Theory, the main ingredients in the proof of our result are the analiticity of the Monomer-Dimer model together with some alternating correlation inequalities.

A recent conjecture of Fyodorov, Hiary & Keating states that the maxima of the Riemann Zeta function on a bounded interval of the critical line behave similarly to the maxima of a specific class of Gaussian fields, the so-called log-correlated Gaussian fields. These include important examples such as branching Brownian motion and the 2D Gaussian free field. In this talk, we will highlight the connections between the number theory problem and the probabilistic models. We will outline the proof of the conjecture in the case of a randomized model of the Zeta function. We will discuss possible approaches to the problem for the function itself. This is joint work with D. Belius (NYU) and A. Harper (Cambridge).

In an ongoing work with Alexander Fribergh and Manuel Cabezas, we give a natural scaling limit for the Random Walk on a simple critical percolation structure, i.e. large critical branching random walks, in high enough dimension. If time permits, I will also describe the scaling limit for the walk projected on the backbone for the incipient infinite cluster and shows how it relates to the specific SSBM, or spatially subordinated brownian motion, which was obtained as a scaling limit in the case of the random walk on the incipient critical tree, in a recent work with M.Cabezas, and formerly introduced as limits of 1d trap models in a joint work with M.Cabezas, R. Royfman and J.Cerny.

Gaussian processes indexed by trees form an interesting class of correlated random fields where the structure of extremal processes can be studied. One popular example is Branching Brownian motion, which has received a lot of attention over the last decades, non the least because of its connection to the KPP equation. In this talk I review the construction of the extremal process of standard and variable speed BBM (with Arguin, Kistler, and Hartung).

Spin glasses are disordered spin systems originated from the desire of understanding the strange magnetic behaviors of certain alloys in physics. As mathematical objects, they are often cited as examples of complex systems and have provided several fascinating structures and conjectures. This talk will be focused on one of the most famous mean-field spin glasses, the Sherrington-Kirkpatrick model. We will present results on the conjectured properties of the Parisi measure including its uniqueness and quantitative behaviors. This is based on joint works with A. Auffinger.

We study an undirected polymer chain living on the one-dimensional integer lattice and carrying i.i.d. random charges. Each self-intersection of the polymer chain contributes an energy to the interaction Hamiltonian that is equal to the product of the charges of the two monomers that meet. The joint probability distribution for the polymer chain and the charges is given by the Gibbs distribution associated with the interaction Hamiltonian. We analyze the annealed free energy per monomer in the limit as the length of the polymer chain tends to infinity.
We derive a spectral representation for the free energy and use this to show that there is a critical curve in the (charge bias, inverse temperature)-plane separating a ballistic phase from a subballistic phase. We show that the phase transition is first order, identify the scaling behaviour of the critical curve for small and for large charge bias, and also identify the scaling behaviour of the free energy for small charge bias and small inverse temperature. In addition, we prove a large deviation principle for the joint law of the empirical speed and the empirical charge, and derive a spectral representation for the associated rate function. This in turn leads to a law of large numbers and a central limit theorem.
What happens in higher dimensions remains largely open. We report on some preliminary results.

Joint work with Q. Berger, F. Caravenna, N. Petrelis and J. Poisat.

We provide a new proof of exponential decay of correlations for subcritical Bernoulli percolation on Z^d. The proof is based on an alternative definition of the critical point. The proof extends to the Ising model and to infinite-range models on infinite locally-finite transitive graphs. It also provides a mean-field lower bound for the explosion of the infinite-cluster density in the supercritical regime.

Joint work with Vincent Tassion.

In this talk I will present some limit theorems for the total spin of ferromagnetic Ising models on random graphs. In such systems two different source of randomness are present: the Ising spins that are random variables distributed according to the Boltzmann-Gibbs measure and the spatial disorder given by the random graph. This is the source of distinction between the "random quenched measure", the "average quenched measure" and the "annealed measure". In the uniqueness regime, the Central Limit Theorem (CLT) can be proven for the general tree-like graphs in the random quenched setting. The average quenched CLT is much more challenging, thus we restrict ourselves to two "configuration models": the model with all vertices having degree 2 (CM(2)) and the case where vertices have either degree 1 or degree 2 (CM(1,2)). In the annealed setting the CLT can be proven for the Ising model on the CM(2), CM(1,2) graphs and also on the Generalized Random Graph. This last model has a finite critical temperature. We will discuss the role of the fluctuation of the vertex degree in the limit normal behaviors.

We review some general properties of statistical mechanics systems leading to Legendre structures. The primary origin of these structure is the second principle of thermodynamics. However, they acquire a very ubiquitous presence in the case of the various models. In particular we discuss the case of multi-species ferromagnetic models, spin glasses and neural networks. The connection with the order parameters is pointed out. In particular we give the interpretation of the Parisi functional order parameter in this frame, and point out some generalizations.

Empirical findings have shown that many real-world networks share fascinating features. Indeed, many real-world networks are small worlds, in the sense that typical distances are much smaller than the size of the network. Further, many real-world networks are scale-free in the sense that there is a high variability in the number of connections of the elements of the networks, making these networks highly inhomogeneous. Such networks are typically modeled using random graphs with power-law degree sequences.
In this lecture, we will investigate the behavior of competition processes on scale-free random graphs with finite-mean, but infinite-variance degrees. Take two vertices uniformly at random, or at either side of an edge chosen uniformly at random, and place an individual of two distinct species at these two vertices. Equip the edges with traversal times, which could be different for the two species. Then let each of the two species invade the graph, such that any other vertex can only be occupied by the species that gets there first. Let the speed of the species be the inverse of the expected traversal times of an edge by that species.
We distinguish two cases. When the traversal times are exponential, we see that one (not necessarily the faster) species will occupy almost all vertices, while the losing species only occupied a bounded number of vertices, i.e., the winner takes it all, the loser's standing small. In particular, no asymptotic coexistence can occur. On the other hand, for deterministic traversal times, the fastest species always gets the majority of the vertices, while the other occupies a subpolynomial number. When the speeds are the same, asymptotic coexistence (in the sense that both species occupy a positive proportion of the vertices) occurs with positive probability.
This lecture is based on joint work with Mia Deijfen, Julia Komjathy and Enrico Baroni, and builds on earlier work with Gerard Hooghiemstra, Shankar Bhamidi and Dmitri Znamenski.

We consider a class of reinforcement processes in which there are n colours of balls and at each iteration of the process one selects some subset of the colours to compete against each other, reinforcing the winner. When defined on a graph this is a toy model for reinforcement in the brain, and it is natural to consider the random subgraph of colours (edges) selected a positive proportion of the time. [Joint with Remco van der Hofstad, Alexey Kuznetsov, and Wioletta Ruszel]

We consider translation-invariant percolation models on $Z^d$ satisfying the finite energy and the FKG properties. We provide explicit upper bounds on the probability of having two distinct clusters going from the endpoints of an edge to distance $n$ (this corresponds to a finite size version of the celebrated Burton-Keane argument proving uniqueness of the infinite-cluster). The proof is based on the generalization of a reverse Poincar\'e inequality for Bernoulli percolation which was derived by Chatterjee and Sen. As a consequence, we show how RSW-type estimates recently obtained by Duminil-Copin, Sidoravicius and Tassion imply upper bounds on the probability of the so-called four-arm event for planar random-cluster models with cluster-weight $q\in [1,4]$.

Joint work with Hugo Duminil-Copin and Yvan Velenik

I will discuss a version of the second moment method which is particularly efficient to analyze the extremes of random fields where multiple scales can be identified. The method emerged from work on the extremes of branching Brownian motion, joint with Louis-Pierre Arguin (CUNY) and Anton Bovier (Bonn), and from a recent work with David Belius (NYU) on the cover time by planar Brownian motion. Time permitting, I will also discuss a procedure of local projections which allows, in a number of models, to generate scales from first principles.

In this talk we considered a monomer-dimer model with independent random weights on the complete graph. Under very general conditions on the randomness one can prove that the quenched pressure density is self-averaging. In the case of i.i.d. monomer random weights we prove that the thermodynamical limit of the quenched pressure is given by the solution of a one-dimensional variational principle. We show that such solution is a smooth function of the dimeric weight. A sketch of the proof, based on a gaussian representation of the partition function, and some open questions will be discussed.

I will discuss a proof of chaos in temperature for even p-spin models which include sufficiently many p-spin interaction terms. The approach is based on a new invariance property for coupled asymptotic Gibbs measures, similar in spirit to the invariance property that appeared in the proof of ultrametricity, used in combination with Talagrand's analogue of Guerra's replica symmetry breaking bound for coupled systems.

It is known that percolation with finite-range symmetric bond occupation probability p(x,y) exhibits mean-field behavior above the upper-critical dimension 6. The situation is the same even if we modify p(x,y) to decay as |x-y|^{-d-a} with a>2. If 0<a<2, then the second moment of p diverges and the situation changes: the upper-critical dimension becomes 3a, and the critical two-point function decays like the Riesz potential, not like the Newtonian potential as in the case of a>2. The marginal case of a=2 has not been studied before, even the asymptotic expression of the random walk's Green function is not known. I will explain the ongoing work (with Lung-Chi Chen) on this marginal case, and outline the proof of the asymptotic expression of the critical two-point function, which turns out to have a log correction to the Newtonian potential, for dimensions above or equal to 6.

Spin glass techniques include finding consequences of self-averaging properties such as stochastic stability, and identifying the distributional limit of overlaps between two configurations drawn from the equilibrium measure. For non-Hermitian random matrices, such as the complex Ginibre ensemble, there are analogous steps. A consequence of self-averaging is a simple limiting formula for all mixed matrix moments. But the overlap between two random eigenvectors is more complicated. Its study was initiated by Chalker and Mehlig, but it is not completed. I will describe this and also some physical applications, such as by Fyodorov and coworkers.

Disordered systems are universally characterized by a high degree of metastability, with the consequence that much of their dynamical behavior can be mathematically recast as a random walk on a "rugged landscape". Many models of such walks treat the landscape as effectively one-dimensional. However, a more accurate representation is that of a landscape in very high dimension. Rigorous results, to be discussed in this talk, demonstrate that the random walk in an appropriate limit maps to invasion percolation, and that this in turn leads to some novel, and often surprising and unexpected, behavior.

We formulate general conditions ensuring that a sequence of multi-linear polynomials of independent random variables (called polynomial chaos expansions) converges to a limiting random variable, given by a Wiener chaos expansion over the d-dimensional white noise. This provides a unified framework to study the continuum and weak disorder scaling limits of statistical mechanics systems that are disorder relevant, including the disordered pinning model with renewal exponent between 1/2 and 1, the (long-range) directed polymer model in dimension 1+1, and the two-dimensional random field Ising model. This gives a new perspective in the study of disorder relevance, and leads to interesting new continuum models. However, this approach breaks down for systems where the disorder is marginally relevant, such as the disordered pinning model with renewal exponent 1/2, and the directed polymer model in dimension 2+1. Nevertheless, we show that a scaling limit still exists, and is furthermore universal across the different models. Joint work with Francesco Caravenna and Nikos Zygouras.

We introduce a class of (2 + 1)-dimensional random growth processes, that can be seen as asymmetric random dynamics of discrete interfaces. Interface configurations correspond to height functions of dimer coverings of the infinite hexagonal or square lattice. “Asymmetric” means that the interface has an average non-zero drift. When the asymmetry parameter p − 1/2 equals zero, the infinite-volume Gibbs measures pi_\rho (with given slope \rho) are stationary and reversible. When p\neq 1/2, \pi_\rho is not reversible any more but, remarkably, it is still stationary. In such stationary states, one finds that the height function at a given point x grows linearly with time t with a non-zero speed, := <(hx(t) − hx(0))> = v t and that the typical fluctuations of Q_x(t) are smaller than any power of t.
For the specific case p = 1 and in the case of the hexagonal lattice, the dynamics coincides with the “anisotropic KPZ growth model” studied by A. Borodin and P. L. Ferrari. For a suitably chosen initial condition (that is very different from the stationary state), they were able to determine the hydrodynamic limit and the interface fluctuations, exploiting the fact that some space-time correlations can be computed exactly.

The jamming transition marks the emergence of rigidity in a system of amorphous and athermal grains. It is characterized by a divergent correlation length of the force-force correlation and non-trivial critical exponents that are independent of spatial dimension, suggesting that a mean field theory can correctly predict their values. I will discuss a mean field approach to the problem based on the exact solution of the hard sphere model in infinite dimension. An unexpected analogy with the Sherrington-Kirkpatrick spin glass model emerges in the solution: as in the SK model, the glassy states turn out to be marginally stable, and are described by a Parisi equation. Marginal stability has a deep impact on the critical properties of the jamming transition and allows one to obtain analytic predictions for the critical exponents. The predictions are consistent with a recently developed scaling theory of the jamming transition, and with numerical simulations. Finally, I will briefly discuss some possible extensions of this approach to other open issues in the theory of glasses.

We reformulate percolation on networks as a message passing process and demonstrate how the resulting equations can be used to calculate, among other things, the size of the percolating cluster and the average cluster size. The calculations are exact for sparse networks when the number of short loops in the network is small, but even on networks with many short loops we find them to be highly accurate when compared with direct numerical simulations. By considering the fixed points of the message passing process, we also show that the percolation threshold on a network with few loops is given by the inverse of the leading eigenvalue of the so-called non-backtracking matrix.