Purdue Probability Seminar, Fall 2020

We establish estimates for the coalescence time of semi-infinite directed geodesics in the planar corner growth model with i.i.d. exponential weights. There are four estimates: upper and lower bounds on the probabilities of both fast and slow coalescence on the correct spatial scale with exponent 3/2. Our proofs utilize a geodesic duality introduced by Pimentel and properties of the increment-stationary last-passage percolation process. For fast coalescence our bounds are new and they have matching optimal exponential order of magnitude. For slow coalescence, we reproduce bounds proved earlier with integrable probability inputs, except that our upper bound misses the optimal order by a logarithmic factor. (Joint work with Timo Seppäläinen)

The large deviation principle (LDP) of the KPZ equation has received a lot of attention from the math and physics community. The problem can be split into two regimes: long time and short time. We will discuss our recent progress in both regimes. This talk will cover a joint work with Promit Ghosal and an ongoing joint work with Li-Cheng Tsai.

In Last Passage Percolation (LPP) one assumes i.i.d. weights on the lattice Z^2. The geodesic from the anti-diagonal h(x)=-x to the point (N,N) is an up-right path starting from h and terminating at (N,N) on which the total weight is maximal. Consider now a cylinder H of width \epsilonN^2/3 and length \epsilon^{3/2-}N centered around the point (N,N) and along the straight line going from the point (0,0) to the point (N,N). The geodesic tree consists of all the geodesics going from h and terminating in the cylinder H. We show that for exponential LPP, for a large class of weights on h(x) and with high probability, the geodesic tree coincides on H with a universal stationary tree. Based on joint works with Marton Balazs, Timo Seppelainen and Patrik Ferrari.

In planar last passage percolation (LPP), every vertex of Z^2 is given an i.i.d. non-negative weight and the weight of directed paths between given points is maximized. Much progress in LPP has occurred in "integrable" models, such as when the vertex weights are exponentially distributed, where exact formulas via representation theoretic connections are available. In such models it is well-known that the upper and lower tails of the point-to-point weight have exponents 3/2 and 3 --- matching the exponents of the GUE Tracy-Widom distribution, the scaling limit of the point-to-point weight. This talk will discuss a geometric route to the same exponents in non-integrable models of LPP under a set of natural assumptions on the limit shape and weak initial tail bounds, but no distributional structure of the vertex weights. Not having access to exact formulas, the arguments rely on understanding the geometry of weight-maximizing paths, the concentration of measure phenomenon, and a natural superadditivity-based bootstrapping procedure. This is based on joint work with Shirshendu Ganguly.

In recent years, significant progress has been made in understanding the stochastic topology of noise. In particular, researchers have looked at how topological features behave when they are based off an increasing number of random points in Euclidean space lying at ever greater distances from the origin. In this talk, we will look at how the Euler characteristic of a filtration of random geometric simplicial complexes behaves when the points that generate them come from two distinct families of extreme value distributions. We will demonstrate a functional strong law of large numbers (FSLLN) for the Euler characteristic process—a summary of how this extremal topology evolves—in each distributional context. Based off of joint work with Takashi Owada.

We study the long time behavior of the total mass of the 2d parabolic Anderson model (PAM) with white noise potential, which is the universal scaling limit of 2d branching random walks in small random environments. There are several known results on the long time behavior of the PAM for more regular potentials, but the 2d white noise is very singular and it requires renormalization techniques. In particular, the Feynman-Kac representation, usually the main tool for deriving asymptotics, breaks down. To overcome this problem we use a measure transform and we introduce a new "partial Feynman-Kac representation“. The new representation is based on a diffusion with distributional drift, and we derive Gaussian heat kernel bounds for such diffusions. Based on joint works with Wolfgang König and Willem van Zuijlen.

We consider the stochastic heat equation on [0,1] with Dirichlet boundary conditions driven by a space-time white noise and a locally Lipschitz drift. We show that the well-known Osgood condition is necessary for the solution to blow-up in finite time, providing the converse of a Theorem by Bonder and Groisman. We also consider the same equation on the whole line and show that the Osgood condition is sufficient for the non-existence of global solutions. Various other extensions are provided; we look at the heat equation with fractional Laplacian, spatial colored noise in $\R^d$, and multiplicative noise. Finally, we give the analogous results for the stochastic wave equation in one dimension. This is a joint work with Mohammud Foondun (University of Strathclyde).

In an excited random walk (ERW), stacks of cookies are placed at each vertex of the integer lattice $\mathbb{Z}$. Before taking a step, the walker eats a cookie which induces a drift in the next step of the walk. Recurrence/transience criteria are known for ERW with finitely many cookies at each site, or infinitely many cookies with special structure (e.g. periodic cookie stacks or cookie stacks generated by a Markov chain). We consider an ERW with infinitely many cookies without these special structures, only assuming that the total drift $\delta$ contained in a cookie stack is finite. In this talk, we will give a criteria for recurrence/transience of the ERW in terms of $\delta$. We will also show that in the critical case $|\delta|=1$, the walk may be either recurrent or transient.

The Gaussian ensembles, originally introduced by Wigner may be generalized to an n-point ensemble called the beta-Hermite ensemble. As with the original ensembles we are interested in studying the local behavior of the eigenvalues. At the edges of the ensemble the rescaled eigenvalues converge to the Airy_beta process which for general beta is characterized as the eigenvalues of a certain random differential operator called the stochastic Airy operator (SAO). In this talk I will give a short introduction to the random matrix models, the Stochastic Airy Operator, and the proof of convergence of the eigenvalues, before introducing another interesting eigenvalue process. This process can be characterized as a limit of eigenvalues of minors of the tridiagonal matrix model associated to the beta-Hermite ensemble as well as the process formed by the eigenvalues of the SAO under a restriction of the domain. This is joint work with Angelica Gonzalez.

We study the directed polymer model for general infinite graphs and random walks. We provide sufficient conditions for the existence or non-existence of phase transitions in terms of properties of the graph and of the random walk. We study in some detail (biased) random walk on various trees including the Galton-Watson trees, and provide a range of other examples that illustrate counter-examples to intuitive extensions of the simple random walk on the lattice. (joint work with Inbar Seroussi and Ofer Zeitouni)

We consider the interacting queuing models introduced by Baccelli and Foss and address te question of when the models have a nontrivial critical value for stability. This is joint work with Zhe Wang.

The field of percolation studies the formation of “giant” connected components in various types of random media. In this talk, we will discuss a higher dimensional analogue of this phenomenon, where instead of connected components, we are looking at k-dimensional cycles — topological objects that describe structures in various dimensions. For example, a 0-cycles is a connected component, a 1-cycle is a loop that surrounds a hole, and a 2-cycle is a surface that encloses a cavity. We focus on a continuum percolation model, where the underlying point process is generated on a compact manifold M. Among all the k-cycles formed at random, we consider as ``giant" those that correspond to one of the k-cycles of M. Similarly to the classical study in percolation theory, our goal is to analyze the phase transitions describing the emergence of these giant cycles. We will also present an unexpected (heuristic) connection to the Euler characteristic. This is joint work with Primoz Skraba from Queen Mary University of London. * this talk doesn't require any background in algebraic topology.