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Monday, 18 July 2022

Philippe Sosoe
Title: Concentration for Interacting Diffusions in the KPZ Class
Abstract:

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.

Daniel Ahlberg
Title: In planar first-passage percolation, non-crossing geodesics tend to coalesce
Abstract:
The study of (infinite) geodesics in first-passage percolation was pioneered by Charles Newman and coworkers in the mid 1990s. Their observation that parallel geodesics tend to coalesce has been central in the development of an ergodic theory for infinite geodesics. In fact, coalescence is typical in the sense that any measurable way to extract a translation invariant family of non-crossing geodesics will result in a coalescing family of geodesics. As an application of this fact, we shall see how one can prove the Benjamini-Kalai-Schramm midpoint problem’.

Tuesday, 19 July 2022

Shirshendu Ganguly
Title: Noise sensitivity and chaos in random planar geometry
Abstract:
The Kardar-Parisi-Zhang (KPZ) universality class refers to a broad family of models of random growth which are believed to exhibit certain common features, such as universal scaling exponents and limiting distributions.

Planar first passage percolation models (FPP), where one considers a distortion of the planar Euclidean metric by random noise are canonical examples of random planar metric spaces expected to lie in this class. A key observable of interest is the geodesic, the (random) shortest path between specified endpoints.

These models can also be viewed as examples of  disordered systems admitting complex energy landscapes. In this formulation, the geodesic is the ground state, the configuration with the lowest energy, lying at the base of the deepest valley.

In this talk we will report recent progress in understanding various geometric features of such energy landscapes for certain variants of FPP known as last passage percolation (LPP) admitting certain key algebraic features, and their manifestations in sensitivity to dynamical perturbations of the underlying noise and associated chaotic behavior.

The talk will be based on joint work with Alan Hammond.
Si Tang
Title: The time constant and limiting shape in high-dimensional first passage percolation
Abstract:

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

Erik Bates
Title: Empirical measures along FPP geodesics
Abstract:

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

Christian Houdré
Title: Limiting Laws in Some Subsequences Problems
Abstract:
I will survey various results on the limiting law of the, properly centered and normalized,  length of the longest common/and or increasing subsequences in random words. These limiting laws range from the Gaussian one to the one of the maximal eigenvalues of some Gaussian random matrices, to less explicit ones.  The survey will end with some open questions.
Chang-Long Yao
Title: Convergence of limit shapes for 2D near-critical first-passage percolation
Abstract:

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

Shuta Nakajima
Title: A variational formula for large deviations in FPP under tail estimates
Abstract:

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

Yuri Bakhtin
Title: Dynamic polymers: invariant measures and ordering by noise.
Abstract:

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.

Eric Cator
Title: Corner growth model on the circle
Abstract:

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

Alexandre Stauffer
Title: Non-equilibrium multi-scale analysis and coexistence in competing first-passage percolation
Abstract:

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).

Barbara Dembin
Title: Coalescence of geodesics and the BKS midpoint problem in first-passage percolation
Abstract:
We consider first-passage percolation on Z^2 with independent and identically distributed weights. Under the assumption that the limit shape has at least 32 extreme points, we prove that geodesics with nearby starting and ending points have significant overlap, coalescing on all but small portions near their endpoints. The statement is quantified, with power-law dependence of the involved quantities on the length of the geodesics.

The result leads to a quantitative resolution of the Benjamini--Kalai--Schramm midpoint problem. It is shown that the probability that a geodesic passes through a given edge is smaller than a power of the distance between that edge and the endpoints of the geodesic.

Joint work with Dor Elboim and Ron Peled.

Wednesday, 27 July 2022

Timo Seppäläinen
Title: Busemann functions in random growth and polymer models
Abstract:

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.

Geoffrey Grimmett
Title: Percolation thresholds
Abstract:

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

Kenneth S. Alexander
Title: Disjointness of geodesics for first passage percolation in the plane
Abstract:

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 $r^\xi$.

Leandro P. R. Pimentel
Title: Integration by Parts and KPZ Two-Point Function
Abstract:

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

Christopher Hoffman
Title: Geodesics in First-Passage Percolation
Abstract:

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.