Purdue Probability Seminar, Spring 2021

First-passage percolation defines a random pseudo-metric on Z^d by attaching to each nearest-neighbor edge of the lattice a non-negative weight. Geodesics are paths which realize the distance between sites. This project considers the question of what the environment looks like on a geodesic through the lens of the empirical distribution on that geodesic when the weights are i.i.d.. We obtain upper and lower tail bounds for the upper and lower tails which quantify and limit the intuitive statement that the typical weight on a geodesic should be small compared to the marginal distribution of an edge weight. Based on joint work-in-progress with Michael Damron, Wai-Kit Lam, and Xiao Shen which was started at the AMS MRC on Spatial Stochastic Models in 2019.

Recently Peltola and Wang introduced the multiple SLE(0) process as the deterministic limit of the random multiple SLE(kappa) curves as kappa goes to zero. They prove this result by means of a ``small kappa’’ large deviations principle, but the limiting curves also turn out to have important geometric characterizations that are independent of their relation to SLE(kappa). In particular, they show that the SLE(0) curves can be generated by a deterministic Loewner evolution driven by multiple points, and the vector field describing the evolution of these points must satisfy a particular system of algebraic equations. We show how to generate solutions to these algebraic equations in two ways: first in terms of the poles and critical points of an associated real rational function, and second via the well-known Caloger-Moser integrable system with particular initial velocities. Although our results are purely deterministic they are again motivated by taking limits of probabilistic constructions, which I will explain.

Many important models in integrable probability (e.g. the KPZ equation, solvable directed polymers, ASEP, stochastic six vertex model) can be embedded into Gibbsian line ensembles. The Gibbs property provides a powerful resampling invariance against Brownian bridges over an arbitrary interval. In this talk, I will explain how to study tightness and path regularity of KPZ line ensemble using this hidden probabilistic structure.

Random walks in cooling random environments (RWCRE) are model of random walks in dynamic random environments where the entire environment is resampled at a fixed sequence of times and is kept fixed in between those resampling times. The sequence of resampling times is called the “cooling sequence.” This model interpolates between that of a simple random walk (where the environment is resampled every step) and random walks in (static) random environments (where the environment is never resampled. In this talk I will focus on the limiting distributions of one-dimensional RWCRE in the regime where the random walk in a static random environment is such that the walk is ballistic (positive speed) and has a s-stable limit with index s in (1,2). Since the two extreme cases (resampling every step or never resampling) have Gaussian and s-stable limiting distributions, we expect that there might be a phase transition from Gaussian to stable limits for certain choices of the cooling sequence. In fact, we prove this to be the case for polynomial cooling sequences and we identify exactly when the phase transition occurs. Moreover, we show that somewhat surprisingly a wide variety of limiting distributions (including mixtures of stable and Gaussian distributions and generalized tempered stable distributions) can be obtained by choosing our cooling sequence appropriately. This talk is based on joint work with Luca Avena and Conrado da Costa.

The symmetric inclusion process (SIP) is an interacting particle system discovered as the dual process of a Markov diffusion that conserves the total energy, it can be tough as the inclusion counterpart of the well-known exclusion process. In this talk, I will present the model with an open boundary, i.e. with two reservoirs which create a flux of particles in the bulk putting the model in a non-equilibrium setting. We will see that via the duality property, we can characterize some correlations of its stationary measure and we will also see what are, in this open setting, the so-called hydrodynamic and hydrostatic limit.
Main references are:
[1] Carinci G., Giardinà C., Giberti C., Redig F. (2013). Duality for stochastic models of transport. Journal of Statistical Physics, 152(4), 657-697.
[2] F. C., Gonçalves P., Sau F. (2020). Symmetric inclusion process with slow boundary: hydrodynamics and hydrostatics. arXiv preprint arXiv:2007.11998.

We develop a dynamical approach to infinite volume polymer measures (IVPM) in random environments. We define polymer dynamics in 1+1 dimension as a stochastic gradient flow, and establish ordering by noise. We prove that, for a fixed asymptotic slope, the polymer dynamics has a unique invariant distribution given by a unique IVPM. Moreover, One Force -- One Solution principle holds.

Can one emulate the simple random walk without actually doing anything random? This talk will be about a deterministic version of random walk called rotor walk, and we will measure its performance in emulating the simple random walk with respect to different parameters, e.g., the shape of the trajectory, number of returns to the origin, etc. In particular, we will see that the number of returns to the origin for the rotor walk can be made equal to the same number for the simple random walk. This resolves a conjecture of Florescu, Ganguly, Levine, and Peres (2014). Parts of this talk will be a joint work with Lila Greco, Lionel Levine, and Peter Li.

Random walks in random environments (RWRE) are well understood in the one-dimensional nearest-neighbor case. A surprising phenomenon is the existence of models where the walk is transient to the right, but with zero limiting velocity. More difficulties are presented by nearest-neighbor RWRE on Z^d for d>1, or RWRE on Z where the nearest-neighbor assumption is removed, and jumps of bounded distance are allowed. Questions about directional transience and limiting speed become much more difficult to answer. C. Sabot and his students have found success studying a special case of RWRE on Z^d, where transition probabilities are drawn in an i.i.d. way from a Dirichlet distribution. In the Dirichlet case, directional transience is characterized, and for d \geq 3, the ballistic regime (positive limiting speed) is also characterized. Inspired by the virtues of the Dirichlet model for the nearest-neighbor case of Z^d, we study random walks in dirichlet random environments on Z with bounded jumps. We characterize directional transience and ballisticity, showing that here, unlike in all previously known cases, ballisticity is governed by two parameters, which correspond to two entirely different ways that a walk may be slowed down.

Suppose H and K are subgroups of a finite group G and consider the H-K double cosets. The uniform distribution on G induces a probability distribution on this space of double cosets. I will discuss several examples, then focus on the case when $G = S_n$ and H and K are parabolic subgroups. The double cosets are contingency tables with fixed row and column sums and the induced distribution is the Fisher-Yates distribution, commonly used in statistical tests of independence. The random transpositions Markov chain on $S_n$ induces a natural Markov chain on contingency tables, for which we can study the eigenvalues and eigenfunctions. Joint work with Persi Diaconis.

In this paper, we study spatial averages for the parabolic Anderson model in the Skorohod sense driven by rough Gaussian noise, which is colored in space and time. We include the case of a fractional noise with Hurst parameters $H_0$ in time and $H_1$ in space, satisfying $H_0 \in (1/2,1)$, $H_1\in (0,1/2)$ and $H_0 + H_1 > 3/4$. Our main result is a functional central limit theorem for the spatial averages. As an important ingredient of our analysis, we present a Feynman-Kac formula that is new for these values of the Hurst parameters.

A classic result by Frieze is that the total weight of the minimum spanning tree (MST) of the uniformly weighted graph converges to zeta(3). Recently, this result was extended to a uniformly weighted simplicial complex, where the role of the MST is played by its higher-dimensional analogue--the Minimum Spanning Acycle (MSA). In this talk, we look at the distribution of weights in this random MSA--both in the bulk and in the extremes. We show that the rescaled empirical distribution of weights in the MSA converges to a measure based on the density of the shadow – the object that generalizes the giant component in higher dimensions. We also show that the shifted extremal weights converge to an inhomogeneous Poisson point process. Our results also apply to the Linial-Meshulam model, where one has access only to a fraction of the potential face weights. This is joint work with Sayan Mukherjee and Gugan Thoppe.