We consider irreducible reversible discrete time Markov chains on a finite state space. Mixing times and hitting times are fundamental parameters of the chain. In this talk, we relate them by showing that the mixing time of the lazy chain is equivalent to the maximum over initial states x and large sets A of the hitting time of A starting from x. As an application, we show that the mixing time on a finite binary tree is robust to bounded change of edge conductances. (joint work with Yuval Peres)
The Airy line ensemble arises in scaling limits of growth models, directed polymers, random matrix theory, tiling problems and non-intersecting line ensembles. This talk will mainly focus on the "non-intersecting Brownian Gibbs property" for this infinite ensemble of lines. Roughly speaking, the measure on lines is invariant under resampling a given curve on an interval according to a Brownian Bridge conditioned to not intersect the above of below labeled curves. This property leads to the proof of a number of previously conjectured results about the top line of this ensemble.
The KPZ line ensemble arises as the scaling limit of a diffusion defined by the Doob-h transform of the quantum Toda lattice Hamiltonian. The top labeled curve of this KPZ ensemble is the fixed time solution to the famous Kardar-Parisi-Zhang stochastic PDE. This talk will also introduce this line ensemble and demonstrate that it too has a "softer" Brownian Gibbs property in which resampled Brownian Bridges may cross the lines above and below, but at exponential energetic cost.
I will discuss recent joint work with Lionel Levine
(Cornell) and Wesley Pegden (NYU). The Abelian Sandpile is a
diffusion process on configurations of chips on the integer lattice on
\Z^d. The stabilized single-source sandpile has a distinctive image
which we now know has a continuum limit. Moreover, we can also
explain the fractal structure of the continuum limit.
The top eigenvalues of finite rank perturbations of large Wigner and
sample covariance matrices are known to exhibit a phase transition.
In recent joint work with Balint Virag we show they have a limit near
the transition, solving an outstanding problem in the real case.
Bypassing joint densities, we identify a continuum operator limit.
The resulting deformations of Tracy-Widom(beta) can be characterized
in terms of a hitting probability of a modified Dyson's Brownian
motion, and in terms of a related linear PDE; both feature beta as a
Random matrix theory is a well developed area of mathematics with strong links to various other areas such as mathematical physics, probability, number theory, combinatorics, to mentioned a few. For a very long time, most results in the theory focus on the global distributions of the eigenvalues or the distribution of the eigenvalues at the edge of the spectrum (for example the Tracy-Widom law). Getting information about eigenvalues inside the bulk of the spectrum seemed out of reach and was done in very special cases, such as for matrices with gaussian entries.
The situation changed substantially in the last few years, due to the works of Erdos et. al. and Tao and the speaker. In this talk, I am going to present a new method developed by Tao and myself. This method is motivated by the Lindenberg replacement method in probability theory and enabled us to get limiting distribution of every single eigenvalue in the spectrum. This gives a new way to attack many long standing problems. For instance, combining our method with recent results of Erdos et. al., we recently proved an old conjecture of Wigner-Dyson-Mehta on correlation functions in its full generality.
Second speaker 5:30-6:30: Vladas Sidoravicius (IMPA)
During the talk I will focus on the connectivity properties of
three models with long (infinite)
range dependencies: Random Interlacements, percolation of the vacant set
rod model and Coordinate percolation. The latter model have polynomial
decay in sub-critical and super-critical regime in dimension 3.
I will explain the nature of this phenomenon and why it is difficult to
handle these models technically. In the second half
of the talk I will present key ideas of the multi-scale analysis which
allows to reach some conclusions. At the end I will discuss
applications and several open problems.
If a random variable X is easier to simulate than to analyze,
one way to estimate its expected value E(X) is to generate
n samples that are distributed according to the law of X
and take their average. If the samples are independent, then
(assuming X has finite variance) the estimate will have typical
error O(1/sqrt(n)). But often one can do better by introducing
appropriate forms of negative dependence. In the toy context of
simulating Markov chains to estimate their absorption probabilities,
I'll describe a scheme that uses maximally anticorrelated
identically distributed Bernoulli random variables (aka rotor
routers) and has typical error O((log n)/n), and a related scheme
with typical error O(1/n). This might seem to be optimal, and
indeed one cannot expect the average (X_1+...+X_n)/n to differ
from its expected value E(X) by less than O(1/n). However, by
using weighted averages that assign X_i less weight when i
is near 1 or n and greater weight when i is near n/2, one can get
estimators for E(X) with typical error significantly smaller than O(1/n).
The methods and ideas are mostly probabilistic and combinatorial. No prior knowledge of rotor-routing or smoothing kernels, and no familiarity with (or fondness for) statistics, will be assumed.
Motivated by percolation theory, we give 3 new minor processes, and their correlation kernels. The first process has to do with GUE with an external source, while the second and third with Wishart and Jacobi with "gaps". These processes, are in distribution equivalent to certain percolation processes.The correlation kernel leads to showing that certain scaling limits of these processes lead to the universal Pearcey process of RMT. The original GUE minor process had to do with the joint distribution of the spectrum of the all the minors of an N by N GUE matrix and has now come up in many models, and so seems canonical in describing certain behavior and hopefully these processes will play a similar role.
It is well-known that the large N limit of the Hermitian one-matrix model
is, for polynomial potentials, an analytic function in the coefficients of the potential whose
power series expansion is a generating function enumerating tessellations of a sphere
by polygonal tiles of given shapes. Since the limiting free energy may be determined analytically by
pushing forward onto the space of eigenvalues, one can use this connection to obtain exact counting formulas for planar maps.
When one tries to repeat this process for the two-matrix model, the reduction to eigenvalues is impeded by a tricky
integral over the unitary group: the Harish-Chandra-Itzykson-Zuber integral. The HCIZ integral itself can be viewed as
the partition function of a Gibbs measure on the unitary group. I will discuss joint work with I. Goulden and M. Guay-Paquet
in which we produce a combinatorial problem solved by the asymptotics of the HCIZ model. This problem is a variant of
the classical Hurwitz problem, which asks for the number of branched covers of the sphere having given singular data. By solving
the combinatorial problem directly, we are able to prove compact convergence of the free energy of the HCIZ model.
We consider the ferromagnetic Potts on typical d-regular graphs, and the independent set model on typical bipartite d-regular graphs,
with graph size tending to infinity. We show that the
replica symmetric (Bethe) prediction applies for
all parameter values in these two models.
In this talk I will describe some of the proof techniques, which will give an indication of the contrast with anti-ferromagnetic Potts model and the independent set model at high fugacity on non-bipartite graphs, where the Bethe prediction is known to fail.
This is joint work with Andrea Montanari, Allan Sly and Nike Sun.
The Schramm-Loewner evolution (SLE) is the canonical model of a
non-crossing conformally invariant random curve, introduced by Oded
Schramm in 1999 as a candidate for the scaling limit of loop erased
random walk and the interfaces in critical percolation. The
development of SLE has been one of the most exciting areas in
probability theory over the last decade because Schramm's curves have
now been shown to arise as the scaling limit of the interfaces of a
number of different discrete models from statistical physics. In this
talk, I will describe how SLE curves can be realized as the flow lines
of a random vector field generated by the Gaussian free field, the
two-time-dimensional analog of Brownian motion, and how this
perspective can be used to resolve a number of open conjectures
regarding the sample path behavior of SLE.
Based on joint work with Scott Sheffield.
In first passage percolation, we place i.i.d. non-negative weights on the nearest-neighbor edges of Z^d and study the induced random metric. A long-standing conjecture gives a relation between two "scaling exponents": one describes the variance of the distance between two points and the other describes the transversal fluctuations of optimizing paths between the same points. In a recent breakthrough work, Sourav Chatterjee proved a version of this conjecture using a strong definition of the exponents. I will discuss work I just completed with Michael Damron, in which we introduce a new and intuitive idea that replaces Chatterjee's main argument and gives an alternative proof of the scaling relation. One advantage of our argument is that it does not require a non-trivial technical assumption of Chatterjee on the weight distribution.
I will present some recent progress on the convergence of rescaled large random quadrangulation -- i.e. a large uniform gluing of squares forming a topological sphere -- towards a continuum object called the Brownian map, which is a universal model for a continuum random surface. I will convey some of the main ideas of the proof, which requires a precise study of geodesics in large quadrangulations and in the limiting space, and in particular, of the locus where these geodesics tend to separate. If time allows I will also mention some questions concerning loop models on random quadrangulations.