Monday, 18 July 2022

We survey results on fluctuations in FPP, with an emphasis on fluctuations of the passage time. A main goal of the lectures is to explain the n/(log n) order bound for the variance in FPP with general edge weight distribution.

Tuesday, 19 July 2022

In this talk, we will survey results on the time constants of high-dimensional first passage percolation on Z^d with general edge weight distribution and their implications on the limiting shape. Starting with the computation for the time constant along the axis direction, we will introduce the main tools used in such computations and how these computations extend to more general directions.

Wednesday, 20 July 2022

We consider the standard i.i.d. first-passage percolation model on Z^d. Our primary interest is in the empirical measures of edge-weights observed along geodesics, say from 0 to [n\xi], where \xi is a fixed unit vector. For various dense families of edge-weight distributions, we prove that these measures converge weakly to a deterministic limit as n tends to infinity. The key tool is a new variational formula for the time constant. In this talk, I will derive this formula and discuss its implications for the convergence of both empirical measures and lengths of geodesics.

Thursday, 21 July 2022

Consider Bernoulli first-passage percolation on the triangular lattice in which sites have 0 and 1 passage times with probability p and 1-p, respectively. First, I will discuss a result in the subcritical regime. Let B(p) be the limit shape in the classical shape theorem, and let L(p) be the correlation length. I will show that the re-scaled limit shape B(p)/L(p) converges to a Euclidean disk, as p tends to p_c from below. The proof relies on the renormalization argument, the scaling limit of near-critical percolation and the construction of the collection of continuum clusters. Then, I will review some recent results and problems in the critical and near-critical cases.

Friday, 22 July 2022

We consider the first passage percolation with identical and independent weight distributions. In this talk, we discuss the upper tail large deviations under a tail assumption on the distributions. We prove that the corresponding rate function is described by the so-called discrete p-Capacity, and we study its asymptotic. The talk is based on joint work with Clement Cosco and Florian Schweiger.

Monday, 25 July 2022

Gibbs measures describing directed polymers in random potential are tightly related to the stochastic Burgers/KPZ/heat equations. One of the basic questions is: do the local interactions of the polymer chain with the random environment and with itself define the macroscopic state uniquely for these models? We establish and explore the connection of this problem with ergodic properties of an infinite-dimensional stochastic gradient flow. Joint work with Hong-Bin Chen and Liying Li.

We will introduce a periodic version of the corner growth model and show that it is a solvable model. We can give stationary measures for the model at a fixed time and for the distribution of the space-time paths, which in this model are up-down paths that form rings. This is joint work with Pablo Ferrari (UBA).

Tuesday, 26 July 2022

We consider a natural random growth process with competition on Z^d called first-passage percolation in a hostile environment, that consists of two first-passage percolation processes FPP_1 and FPP_\lambda that compete for the occupancy of sites. Initially FPP_1 occupies the origin and spreads through the edges of Z^d at rate 1, while FPP_\lambda is initialised at sites called seeds that are distributed according to a product of Bernoulli measures of parameter p. A seed remainsdormant until FPP_1 or FPP_\lambda attempts to occupy it, after which it spreads through the edges of Z^d at rate \lambda. We will discuss the results known for this model and present a recent proof that the two types can coexist (concurrently produce an infinite cluster) on Z^d. We remark that, though counterintuitive, the above model is not monotone in the sense that adding a seed of FPP_\lambda could favor FPP_1. A central contribution of our work is the development of a novel multi-scale analysis to analyze this model, which we call a multi-scale analysis with non-equilibrium feedback and which we believe could help analyze other models with non-equilibrium dynamics and lack of monotonicity. Based on a joint work with Tom Finn (Univ. of Bath).

Wednesday, 27 July 2022

Understanding the random geometric properties of stochastic growth models is a current challenge in probability theory. Busemann functions, a notion adapted from metric geometry, have emerged as a useful analytic tool to study features of random growth. This talk describes recent progress in constructing and describing the Busemann process of several models, possibly including the corner growth model, Brownian last-passage percolation, directed polymers, the KPZ equation, and the directed landscape. Studying the Busemann process leads to results on the uniqueness and coalescence structure of semi-infinite geodesics, extrema of variational formulas of limit shapes, one-force-one-solution principles, and the non-existence of bi-infinite geodesics and bi-infinite polymer measures. This talk is based on collaborations with M. Balazs, E. Bates, O. Busani, W. L. Fan, C. Janjigian, F. Rassoul-Agha, and E. Sorensen.

In classical percolation theory, one studies the geometry of a random subset of a Euclidean graph (such as the square lattice). The key questions concern the existence and nature of percolation phase transitions. This lecture is an account of rigorous work from 1960 to 2022 on exact values of, and relations between, the values of critical points for percolation on planar graphs. The special role of isoradial graphs will be exposed. Certain predictions (from 1964) of Sykes and Essam concerning matching pairs of lattices can be extended to hyperbolic space (joint work with Zhongyang Li).

Thursday, 28 July 2022

For first passage percolation in the plane, we consider the probability of disjointness for pairs of approximately parallel geodesics. We work in the context of a special type of isotropic random graph, chosen to be as ``lattice—like’’ as possible, in order to exploit rotational symmetry. Consider two points $x,p$ above the horizontal axis, their mirror images $y,q$ below the axis, and the (approximately horizontal) geodesics $\Gamma_{xp}$ and $\Gamma_{yq}$. Suppose the vertical separations are $|y-x|=a,|q-p|=b$ with $b\geq a$, and the horizontal separation of the two pairs is $L$. We show that the probability that $\Gamma_{xp}$ and $\Gamma_{yq}$ are disjoint behaves roughly as $a^{1/\xi-1}b/L$, to within subpolynomial corrections, where $\xi$ is the wandering exponent (believed to be 2/3) such that a geodesic of length $r$ typically wanders by order $r^\xi$ from the straight line connecting its endpoints. (We have to assume the existence of such a $\xi$, in a certain natural sense, since it is not proven.) Along the way we prove a ``small ball'' result for the probability a geodesic wanders transversally much less than $r^\xi$.

We apply the integration by parts formula from Malliavin calculus to study establish a key relation between the two-point correlation function of the Kardar-Parisi-Zhang (KPZ) Fixed Point and the location of the maximum of an Airy process plus a Brownian motion with a negative parabolic drift. Integration by parts also allows us to deduce the density of this location in terms of the second derivative of the variance of the KPZ fixed point. We further develop an adaptation of Malliavin-Stein method that implies asymptotic independence with respect to the initial data.

Friday, 29 July 2022

I will discuss several results and conjectures about geodesics in first-passage percolation. I will also discuss how these relate to conjectures about the limiting shape.