**Minicourses**

Marek Biskup: *A random walk driven by the two-dimensional discrete Gaussian Free Field. *I will describe the long time behavior of a random walk in random environment that is derived from samples of the two-dimensional Discrete Gaussian Free Field (DGFF). Perhaps the best way to describe the problem is in physics context: Assume the walker carries a unit electric charge and that the DGFF represents the electrostatic potential; the walker thus feels a drift in the directions where the electric field increases. I will show various aspects of subdiffusive behavior that result from the interaction including explicit exponents as well as a phase transition in the strength of the interaction. Notwithstanding, the main emphasis of the lectures will be on the techniques needed in proofs. These include various duality arguments for random resistor networks, concentration of measure as well as sharp-threshold arguments from two-dimensional near-critical percolation. Based on joint work with J. Ding and S. Goswami.

Jonathan Mattingly: *The anatomy of some ergodic theorems in stochastic dynamics. *Through examples, I will explore the central ideas in a number of ergodic theorems in stochastic dynamics. Examples will largely be drawn from stochastic ODEs and PDEs.

**Talks**

Louis-Pierre Arguin: *The maximum of the Riemann zeta function in a short interval of the critical line**. *A conjecture of Fyodorov, Hiary & Keating states that the maxima of the modulus of the Riemann zeta function on an interval of the critical line behave similarly to the maxima of a log-correlated process. In this talk, we will discuss a proof of this conjecture to leading order, unconditionally on the Riemann Hypothesis. We will highlight the connections between the number theory problem and the probabilistic models including the branching random walk. We will also discuss the relations with the freezing transition for this problem. This is joint work with D. Belius (Zurich), P. Bourgade (NYU), M. Radizwill (McGill), and K. Soundararajan (Stanford).

Rodrigo Bañuelos: *On the p-norm of the discrete Hilbert transform. *The discrete Hilbert transform was introduced by David Hilbert at the beginning of the 20th century. In 1925, Marcel Riesz proved the L^p boundedness, 1<p<infty, of the continuous version of this operator (also called the conjugate function). From this, he deduced the same result for the discrete version. In 1926, Titchmarsh gave a new proof of Riesz’s result and claimed that in addition the two operators have the same p-norms. The following year, Titchmarsh pointed out that his argument for the equality of the norms was incorrect. The problem of equality has been a long-standing conjecture since. In this talk we describe a proof of this conjecture based on martingale inequalities and Doob-h transforms. This is joint work with Mateusz Kwasnicki of Wroclaw University.

Fabrice Baudoin: *Hypocoercive diffusion processes and gradient bounds. *We will study several methods, both probabilistic and analytic, yielding gradient bounds for the semigroups associated to Kolmogorov type operators. Some of those bounds will be shown to imply convergence to equilibrium with explicit rates. Several models will be discussed in details. The talk will be based on joint works with Camille Tardif, Maria Gordina and Phanuel Mariano.

Gerandy Brito: *Absence of infinite backward paths in first passage* *percolation in any dimension.* We focus on the geometry of first passage percolation in higher dimensions and analyze the structure of infinite geodesics, which are infinite paths with the property that any finite segment is a geodesic. We prove that backward paths are almost surely finite for dimension d ≥ 2 in geodesic graphs obtained as weak limit of geodesics to hyperplanes. Our result suggests that bigeodesics cannot be constructed in any dimension by taking limits of geodesics to hyperplanes, and this relates to the nonexistence of bigeodesics, a conjecture which is open even in d = 2. This is based on joint work with Michael Damron and Jack Hanson.

Shirshendu Ganguly: *Polymer geometry in the large deviation regime. *Directed last-passage percolation models in the plane, where one studies the weight and the geometry of the highest weight paths (also referred to as polymers) between two far away vertices in a field of i.i.d. weights, provide paradigm examples of random interfaces believed to exhibit the Kardar-Parisi-Zhang (KPZ) universality. One important object of study is the scaling of the transversal fluctuation of a polymer between (0, 0) and (n, n), i.e., the maximum distance between a polymer and the diagonal line joining the end points. It is widely believed that the transversal fluctuation exhibits the characteristic KPZ scaling exponent of 2/3, although this is rigorously known only for a few exactly solvable models. In this talk, I will discuss the question of polymer geometry in the large deviation regime, i.e., when the weight of the polymer is conditioned to be either macroscopically larger (upper tail) or smaller (lower tail) than typical. Precise asymptotics of the large deviation probabilities are known only for a few integrable models. Moreover, for Poissonian last-passage percolation, it was shown by Deuschel and Zeitouni (1999) that in the upper tail regime the polymers continue to concentrate around the diagonal while the behavior in the lower tail regime was left open. Resolving the latter, I will discuss a result showing that the polymer does not concentrate around any deterministic curve in the lower tail regime. Our proof does not use integrability and works for a large class of last-passage percolation models. Based on joint work with Riddhipratim Basu and Allan Sly.

Jack Hanson: *Half-space critical exponents in high-dimensional percolation. *Critical percolation is fairly well-understood on Z^d for d > 11. Exact values of many critical exponents are rigorously known: for instance, the “one-arm” probability that the origin is connected by an open path to distance r scales as r^{-2}. However, most existing methods rely heavily on the symmetries of the lattice, and do not extend to fractional spaces. We will discuss progress on these questions in the high-dimensional upper half-space, including a proof that the half-space one-arm probability scales as r^{-3}.

David Herzog: *Ergodicity and Lyapunov functions for Langevin dynamics with singular potentials. *We discuss Langevin dynamics of N particles on R^d interacting through a singular repulsive potential, e.g. the well-known Lennard-Jones type, and show that the system converges to the unique invariant Gibbs measure exponentially fast in a weighted total variation distance. The proof of the result turns on an explicit construction of a Lyapunov function. In contrast to previous results for such systems, our result implies geometric convergence to equilibrium starting from an essentially optimal family of initial distributions.

Jessica Lin: *Stochastic Homogenization for Reaction-Diffusion Equations.* I will present several results concerning the stochastic homogenization for reaction-diffusion equations. We consider heterogeneous reaction-diffusion equations with stationary and ergodic nonlinear reaction terms. Under certain hypotheses on the environment, we show that the typical large-time, large-scale behavior of solutions is governed by a deterministic front propagation. Our arguments rely on analyzing a suitable analogue of “first passage time” for solutions of reaction-diffusion equations. In particular, under these hypotheses, solutions of heterogeneous reaction-diffusion equations with front-like initial data become asymptotically front-like with a deterministic speed. This talk is based on joint work with Andrej Zlatos.

Firas Rassoul-Agha: *Shifted weights and restricted-length paths in* *first-passage percolation. *We study standard first-passage percolation via related optimization problems that restrict path length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about the convergence of geodesic length due to Hammersley, Smythe and Wierman, and Kesten. We study the regularity of the time constant as a function of the shift of weights. For unbounded weights, this function is strictly concave and in case of two or more atoms it has a dense set of singularities. For any weight distribution with an atom at the origin there is a singularity at zero, generalizing a result of Steele and Zhang for Bernoulli FPP. The regularity results are proved by the van den Berg-Kesten modification argument. Joint work with Arjun Krishnan and Timo Seppalainen.

Philippe Sosoe: *Phase transition in 1 dimensional Gibbs measure. *In 1980s, Lebowitz, Rose and Speer introduced a measure of Gibbs type based on the Gaussian free field (a type of Brownian bridge) on the circle, which they argued should be invariant for the deterministic nonlinear Schroedinger equation. They also identified a phase transition: their construction depends on a certain cutoff, above which the measure no longer exists. I will explain their result, as well as recent work concerning what happens at the critical value for the cutoff.

**Short Talks**

Erik Bates: *Low-temperature localization of directed polymers. *On the d-dimensional integer lattice, directed polymers are paths of a random walk that have been reweighted according to a random environment that refreshes at each time step. The qualitative behavior of the system is governed by a temperature parameter. If this parameter is small, the environment has little effect, meaning all possible paths are close to equally likely. If the parameter is made large, however, the system undergoes a phase transition at which the path’s endpoint starts to localize. I will describe some recent results that, by examining the localization at a global level, demonstrate just how dramatic this phase transition is. (joint work with Sourav Chatterjee)

Ziteng Cheng: *Passage times of an additive functional driven by a time-inhomogeneous Markov chain : a Wiener-Hopf approach. *An additive functional driven by a Markov chain is a process whose future increment depends only on the current state of the driving Markov chain. Despite the accessibility of such definition, the joint distribution of the passage time and the driving Markov chain at that time is difficult to obtain. It was shown that, when the driving Markov chain is time-homogeneous and the additive functional is of a specific type, the Laplace transform of the joint distribution is the unique solution of a certain matrix equation. Such result is called Wiener-Hopf factorization for time homogeneous Markov chains. We generalize the previous result to the case of time-inhomogeneous Markov chains and the corresponding Wiener-Hopf factorization is expressed in term of an operator equation. The obtained result has applications in fluid models and will prepare the foundation for studying the corresponding inverse problems. This is joint work with Tomasz R. Bielecki, Igor Cialenco and Ruoting Gong.

Daesung Kim: *Stability and instability results for the log Sobolev inequality. *The logarithmic Sobolev inequality states that the Fisher information is bounded below by the relative entropy. Equality holds if and only if a measure is Gaussian. We are interested in measuring how far a measure is away from Gaussian measures when it is close to achieving the equality. To this end, we find a lower bound of the deficit, which is the difference between the Fisher information and the relative entropy, in terms of distances. In this talk, we discuss deficit bounds for the log Sobolev inequality in terms of the Wasserstein distances and $L^{1}$ distance. We also show that these results are best possible by giving an explicit example. This is based on a joint work with Emanuel Indrei.

Jinsu Kim: *Stochastically modeled reaction networks: positive recurrence and mixing times. *Reaction networks are graphical configurations that can be used to describe biological interaction networks. If the abundances of the constituent species of the system are low, we can model the dynamics of species counts in a jump by jump fashion as a continuous time Markov chain. In this talk, we will mainly focus on which conditions of the graph imply positive recurrence (existence of a stationary distribution) for the associated continuous time Markov chain. I will also present results related to their mixing times, which give the time required for the distribution of the continuous Markov chain to get close to the stationary distribution.

Phanuel Mariano: *Gradient Bounds for Kolmogorov Type Diffusions. *The Kolmogorov diffusion is the joint process of Brownian motion together with integrated Brownian motion, that is X_t=(B_{t},\int_0^t B_s ds). The generator of this process is known to be hypoelliptic (diffusions driven by vector fields whose Lie algebra span the whole tangent space) rather than elliptic. We study gradient bounds and other functional inequalities for the diffusion semigroup generated by Kolmogorov type operators. The focus is on two different methods: coupling techniques and generalized Gamma-calculus techniques. For the coupling technique, we use a coupling by parallel translation to induce a coupling on the Kolmogorov type diffusions. In the Gamma-calculus approach, we will prove a new generalized curvature dimension inequality to study various functional inequalities. The class of processes we study are general and includes Kolmogorov diffusions where the Brownian motion lives on a Riemannian manifold. This talk is based on joint work with Fabrice Baudoin and Maria Gordina.

Marcus Michelen: *Central limit theorems from the roots of probability generating functions. *For each n, let X_n in {0,…,n} be a random variable, with mean mu_n and standard deviation sigma_n, and let P_n(z) be its probability generating function. We show that if none of the complex zeros of the polynomials {P_n(z)} are contained in a neighbourhood of 1 in C and sigma_n > n^{epsilon}, for some epsilon >0, then X_n^* =(X_n – mu_n) sigma^{-1}_n tends to a normal random variable in distribution as n to infty. This result is sharp in the sense that there exist sequences of random variables {X_n} with sigma_n > Clog n for which P_n(z) has no roots near $1$ and X_n^* is not asymptotically normal. These results disprove a conjecture of Pemantle and improve upon various results in the literature. We go on to prove several other results connecting the location of the zeros of P_n(z) and the distribution of the random variables X_n. This is based on joint work with Julian Sahasrabudhe.

Josh Rosenberg: *Quenched survival of Bernoulli percolation on Galton-Watson trees. *In this talk I will explore the subject of Bernoulli percolation on Galton-Watson trees. Letting g(T,p) represent the probability a tree T survives Bernoulli percolation with parameter p, we establish several results relating to the behavior of g in the supercritical region. These include an expression for the third order Taylor expansion of g at criticality in terms of limits of martingales depending on T, a proof that g is smooth in the supercritical region, and a proof that g’ extends continuously to the boundary of the supercritical region. Allowing for some mild moment constraints on the offspring distribution, each of these results is shown to hold for almost every Galton-Watson tree. This is based on joint work with Marcus Michelen and Robin Pemantle.

Pengfei Tang: *Heavy Bernoulli percolation clusters are indistinguishable. *We prove that for Bernoulli percolation on quasi-transitive nonunimodular graphs heavy clusters are indistinguishable. As an application, we also show that the uniqueness threshold is also the threshold for connectivity decay for general quasi-transitive graphs.

Fanhui Xu: *Function spaces of generalized smoothness and the Cauchy* *problem of parabolic integro-differential equations therein. *A scaling function that describes a class of scalable Levy measures is introduced, and is used to define generalized Holder smoothness. Embedding relations among the generalized Holder space and some other auxiliary spaces are established by utilizing probabilistic representations. The Cauchy problem of parabolic Kolmogorov equations is studied in these function spaces of generalized smoothness. Existence and uniqueness of a solution is proved by deriving apriori estimates. Regularity results are obtained simultaneously. They preserve the shape of estimates in standard Holder classes and extend the existing results of Cauchy problems associated to alpha-stable processes.

Jiayan Ye: *Stationary DLA is equivalent to DLA of a long line.* I will discuss the stationary harmonic measure on the upper half plane and the corresponding diffusion limit aggregation (DLA). We prove that the stationary harmonic measure is equivalent to another measure. As an application, we prove that the stationary harmonic measure is equivalent microscopically to the standard harmonic measure of a long line. This is a joint work with Eviatar Procaccia and Yuan Zhang.