ColumbiaPrinceton Probability Day 2017
Friday, March 31, 2017 • Princeton University 
Title: Mathematics of multiparticle diffusion limited aggregation Abstract: In late seventies H. Rosenstock and C. Marquardt introduced the following stochastic aggregation model on Z^d: Start with particles distributed according to the product Bernoulli measure with parameter μ. In addition, start with a static aggregate at the origin. Nonaggregated particles move as continuoustime simple random walks obeying the exclusion rule, whereas aggregated particles do not move. The aggregate grows by attaching particles to its surface whenever a particle attempts to jump onto it. This evolution is called Multiparticle Diffusion Limited Aggregation (MDLA). Its difficult to analyze structure inspired Witten and Sander to introduce in 1981 a "simplified version" of the model with only one particle moving at a step of procedure, which became well known as celebrated DLA model. MDLA model even in dimension 1 has highly nontrivial behavior. In its independent random walk version as well as for exclusion process H. Kesten and V. Sidoravicius proved that if the original density of particles is smaller than one, then the aggregate is growing sub linearly. More than ten years later A. Sly showed that for the density larger than one it advances linearly, establishing a remarkable phase transition. In my talk I will briefly review known results and will focus on the progress in dimensions d>1. For independent random walk systems A. Sly argument implies linear growth of the farmost reaching arm of the aggregate for the range of densities bounded away from zero. Our main result (joint with A. Stauffer) states that for the exclusion version of the process if d>1 and the initial density of particles is large enough, then with positive probability the aggregate has linearly growing arms. In fact it obeys a certain type of the shape theorem. The key conceptual element of our analysis is the introduction and study of a new growth process. Study of this process strongly indicates that high density MDLA belongs to KardarParisiZhang (KPZ) universality class. Very intriguing question is if in dimension d>1 there exists similar phase transition as in dimension one, and how it affects formation of geometric shapes of the aggregate. In other words is MDLA undergoing transition from DLA type growth (at low density) to KPZ type growth (for high density).

Title: Invertibility and condition number of sparse random
matrices Abstract: Consider an n by n linear system Ax=b. If the righthand side of the system is known up to a certain error, then in process of the solution, this error gets amplified by the condition number of the matrix A, i.e. by the ratio of its largest and smallest singular values. This observation led von Neumann and his collaborators to consider the condition number of a random matrix and conjecture that it should be of order n. This conjecture of von Neumann was proved in full generality a few years ago. In this talk, we will discus whether von Neumann's conjecture can be extended to sparse random matrices. We will also discus invertibility of the adjacency matrix of a directed ErdosRenyi graph. Joint work with Anirban Basak.

Title: Random walk driven by twodimensional discrete Gaussian
free field Abstract: I will discuss recent progress on the understanding of the random walk driven by twodimensional discrete Gaussian Free Field (DGFF). Specifically, I will consider the walk that jumps an edge of the square lattice with probability proportional to the exponential of the gradient of DGFF across that edge. The walk thus tends to move in the direction of the gradient of the DGFF and this results in trapping. I will demonstrate this effect by showing a kind of subdiffusive behavior with explicit (subdiffusive) exponents that are in agreement with conjectures from physics. The method of proof is interesting in its own right as it is based on a version of the RussoSeymourWelsh Theorem for effective resistance naturally associated with this problem. Based on recent joint work with Jian Ding and Subhajit Goswami.

Title: Laguerre and Jacobi analogues of the Warren process Abstract: We define Laguerre and Jacobi analogues of the Warren process. That is, we construct local dynamics on a triangular array of particles so that the projections to each level recover the Laguerre and Jacobi eigenvalue processes of KönigO'Connell and Doumerc and the fixed time distributions recover the joint distribution of eigenvalues in multilevel Laguerre and Jacobi random matrix ensembles. Our techniques extend and generalize the framework of intertwining diffusions developed by PalShkolnikov. One consequence is the construction of particle systems with local interactions whose fixed time distribution recovers the hard edge of random matrix theory.

Title: Discrete interface dynamics and hydrodynamic limits Abstract: Dimer models provide natural models of (2+1)dimensional random discrete interfaces and of stochastic interface dynamics. I will discuss two examples of such dynamics, a reversible one and a driven one (growth process). In both cases we can prove the convergence of the stochastic interface evolution to a deterministic PDE after suitable (diffusive or hyperbolic respectively in the two cases) spacetime rescaling. Joint work with B. Laslier and M. Legras.

Title: Local regime of 1d random band matrices Abstract: Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between meanfield type Wigner matrices (where all matrix elements are independent up to the symmetry) and random Schrodinger operators. However, after several decades of intensive research, many of the spectral properties of RBM are still not understood on a mathematical level of rigour, even in a one dimensional case. In this talk we will discuss an application of the supersymmetric method (SUSY) to the analysis of the bulk local regime of some specific types of RBM. We present rigorous results about the crossover for 1d RBM on the level of characteristic polynomials, as well as some progress in studying of the density of states and usual second correlation function. 