Columbia-Princeton Probability Day 2016

Columbia University, April 8, 2016

Confirmed Speakers
  • Guillaume Barraquand (Columbia)
  • Rick Kenyon (Brown)
  • Joel Lebowitz (Rutgers)
  • Hao Shen (Columbia)
  • Allan Sly (Berkeley)
  • Peter Winkler (Dartmouth)

Schedule (tbc)

9-9:30: registration (508 Math)
9:30-10:30: Winkler (203 Math)
10:30-11:30: Kenyon
11:30-12:00 break
12:00-1:00: Sly
1:00-2:30: lunch
2:30-3:30: Lebowitz
3:30-4: break
4-4:30: Barraquand
4:30-5: Shen

Practical Information
The conference will take place in Columbia University's Mathematics Building Room 203, on April 8, 2016. (Breaks in 508 Math).

Directions to Columbia's Mornigside Heights Campus.

Please register in order to attend the conference. (Registration is free but kindly requested).

Titles and Abstracts

  • Peter Winkler
    Permutation Limits and Pattern Densities

    The space of permutations of 1,2,...,n for all n has a natural completion to the space of probability measures on the unit square with uniform marginals (also known as doubly-stochastic measures, or 2-dimensional copulas).  The "pattern density" of a fixed permutation in a copula is the probability that the relative orders of the x- and y-coordinates of random points yield that permutation.
           We want to know not only which sets of pattern densities are realizable, but what most permutations with given densities look like.  To do this we employ a variational principle saying that the copula representing most members of such a class of permutations is the one maximizing a certain entropy integral.  
           In work done last spring at ICERM with Rick Kenyon (Brown), Dan Kral' (Warwick) and Charles Radin (Texas), analysis and heuristics have been used to make a dent in this large set of problems.

  • Rick Kenyon
    Dimers and geometry

    Given a 3-connected planar graph G, the space of embeddings with  convex faces and pinned boundary is homeomorphic to a ball. We give global coordinates and natural probability measures on this space.

    We also discuss various subspaces of embeddings of planar graphs, including those with fixed face shapes, fixed edge directions, fixed face areas, and the relation of all of these concepts with the Kasteleyn matrix.

  • Allan Sly
    Counting Solutions to Random Constraint Satisfaction Problems

    Satisfaction problems subject to random constraints are a well-studied area in combinatorics and the theory of computation, for example random colourings of random graphs and random k-sat.  Ideas from statistical physics provide a detailed description of phase transitions and properties of these models.  I will discuss the condensation regime where these model undergo a one step replica symmetry breaking transition.

    Joint work with Nike Sun and Yumeng Zhang

  • Joel Lebowitz
    Fluctuations, Large Deviations and Rigidity in Superhomogeneous Systems

    Superhomogeneous (a.k.a. hyperuniform) particle systems are point processes on R^d (or Z^d) that are translation invariant (or periodic) and for which the variance of the number of particles in a region V grows slower than the volume of V. Examples include Coulomb systems, determinantal processes with projection kernels and certain perturbed lattice models. I will first review some old work on superhomogeneous systems and and then describe some new work (with Subhro Ghosh) providing sufficient conditions (involving decay of pair correlations) for number rigidity in such systems in dimension d=1,2. A particle system is said to exhibit number rigidity if the probability distribution of the number of particles in a bounded region R, conditioned on the particle configuration in R^c, is concentrated on a single integer N. All known (to us) examples in which number rigidity has been established in d=1,2 satisfy our conditions, and we conjecture that superhomogeneity is also a necessary condition for number rigidity in all dimensions d. On the other hand, it follows from the work of Peres and Sly on perturbed lattice systems that in d>2 there are no such sufficiency conditions involving the decay of correlations.

  • Guillaume Barraquand
    Random walks in Beta random environment.

    We consider a model of random walks in space-time random environment, with Beta-distributed transition probabilities. This model is exactly solvable, in the sense that the law of the (finite time) position of the walker can be completely characterized by Fredholm determinantal formulas. This enables to prove a limit theorem towards the Tracy-Widom distribution for the second order corrections to the large deviation principle satisfied by the walker, thus extending the scope of KPZ universality to RWRE. We will also present a few similar results about degenerations of the model: a first passage percolation model which is the "zero-temperature" limit, and a certain diffusive limit which leads to well-studied stochastic flows. (Joint work with Ivan Corwin)

  • Hao Shen
    Regularity structure theory and its applications

    I will review the basic ideas of the regularity structure theory recently developed by Martin Hairer, as well as its applications to stochastic PDE problems. I will then discuss my joint works with Hairer on well-posedness of the sine-Gordon equation, and central limit theorems for stochastic PDEs.