Fall 2014 :
Spring 2014 :
Fall 2013 :
Spring 2013 :
Fall 2012 :
Spring 2012 :
Fall 2011 :
Spring 2011 :
Fall 2010 :
Spring 2010 :
Fall 2009 :
Spring 2009 :
Semester Programs :
MIT Probability Seminar, Spring 2015
Monday 4.15 - 5.15 pm
the MIT probability seminar to your Google calendar.
February 9 February 17
(Harvard) Change of date and venue due to storm on the 7th. Room E17-122.
On the chemical distance in critical percolation
In two-dimensional critical percolation, the works of
Aizenman-Burchard and Kesten-Zhang imply that macroscopic distances
inside percolation clusters are bounded below by a power of the
Euclidean distance greater than 1+ε, for some positive
ε. No more precise lower bound has been given so
far. Conditional on the existence of an open crossing of a box of side
length n, there is a distinguished open path which can be
characterized in terms of arm exponents: the lowest open path crossing
the box. This clearly gives an upper bound for the shortest path. The
lowest crossing was shown by Zhang and Morrow to have volume
n4/3+o(1) on the triangular lattice.
Following a question of Kesten and Zhang, we compare the length of
shortest circuit in an annulus to that of the innermost circuit
(defined analogously to the lowest crossing). I will explain how to
show that the ratio of the expected length of the shortest circuit to
the expected length of the innermost crossing tends to zero as the
size of the annulus grows.
Joint work with Jack Hanson and Michael Damron.
Alexander Holroyd (MSR)
Finitely dependent coloring
A central concept of probability and ergodic theory is mixing in its various forms. The strongest and simplest mixing condition is finite dependence, which states that variables at sufficiently well separated locations are independent. A 50-year old conundrum is to understand the relationship between finitely dependent processes and block factors (a block factor is a finite-range function of an independent family). The issue takes a very surprising new turn if we in addition impose a local constraint (such as proper coloring) on the process. In particular, this has led to the discovery of a beautiful yet mysterious stochastic process that seemingly has no right to exist.
Nick Gravin (MSR)
Towards Optimal Algorithms for Prediction with Expert Advice
We study the classical problem of prediction with expert advice in the adversarial setting with a geometric stopping time. In 1965, Cover gave the optimal algorithm for the case of 2 experts. In this work, we design the optimal algorithm, adversary and regret for the case of 3 experts. Further, we show that the optimal algorithm for 2 and 3 experts is a probability matching algorithm (analogous to Thompson sampling) against a particular randomized adversary. It turns out that this algorithm is not only optimal against this adversary, but also minimax optimal against all possible adversaries.
We establish a constant factor separation between the regrets achieved by the optimal algorithm and the widely used multiplicative weights algorithm. Along the way, we improve the regret lower bounds for the multiplicative weights algorithm for an arbitrary number of experts and show that this is tight for 2 experts. In our analysis, we develop upper and lower bounds simultaneously, analogous to the primal-dual method. The analysis of the optimal adversary relies on delicate random walk estimates. We further use this connection to develop an improved regret bound for the case of 4 experts, and provide a general framework for designing the optimal algorithm for an arbitrary number of experts.
This is a joint work with: Yuval Peres, Balasubramanian Sivan
Senya Shlosman (Marseille)
Ising random walks and their pinning properties
Ising random walk is a random walk with certain self-interaction. It describes the behavior of the interface, separating the (+)-phase from the (-)-phase of the 2D Ising model. One says that pinning is taking place, when, due to the self-interaction, the qualitative behavior of the random walk is different from the non-interacting case. I will explain when pinning does happen and when it does not happen.
Based on a joint work with D. Ioffe and F. Toninelli, arXiv:1407.3592
March 23 Spring vacation
Boris Hanin (MIT)
Pairing Between Zeros and Critical Point of Random Polynomials
Let pN be a degree N polynomial in one complex
variable. The purpose of this talk is to explain why the zeros and
critical points of pN come in pairs, spaced about
N-1 apart. I will explain how such a pairing arises by
relating zeros and critical points to electrostatics on the Riemann
Nikeghbali (University of Zurich)
Ratios in random matrix theory and number theory
We shall consider the Circular Unitary Ensemble and
ratios of characteristic polynomials at the microscopic
scale in this ensemble. These ratios have been
extensively studied in the past decade, using various
techniques and approaches, in problems related for
instance to mathematical physics or number theory (value
distribution of the Riemann zeta function). It has been
observed by various authors that the average of these
ratios and their logarithmic derivatives converge (at
the microscopic scale) when the size of the unitary
group goes to infinity. In this talks we shall provide a
probabilistic framework from which we deduce that the
ratios themselves converge to some explicit random
rational function and we compute explicitly the average
of this object. We also explain how we expect these
limiting objects to be related to corresponding ratios
for the Riemann zeta function.
Thursday April 9, 3 to 4pm, room 66-168
A quantitative Burton-Keane estimate under strong FKG condition
Abstract: We consider translation-invariant percolation models on
Zd satisfying the finite energy and the FKG properties. We
provide explicit upper bounds on the probability of having two
distinct clusters going from the endpoints of an edge to distance n
(this corresponds to a finite size version of the celebrated
Burton-Keane argument proving uniqueness of the infinite-cluster). The
proof is based on the generalization of a reverse Poincare
inequality for Bernoulli percolation which was derived by Chatterjee
and Sen. As a consequence, we show how RSW-type estimates recently
obtained by Duminil-Copin, Sidoravicius and Tassion imply upper bounds
on the probability of the so-called four-arm event for planar
random-cluster models with cluster-weight q in [1,4].
Joint work with Hugo Duminil-Copin and Yvan Velenik
Kavita Ramanan (Brown University)
Obliquely reflected diffusions in rough planar domains
Obliquely reflected diffusions in smooth domains are classical objects
that have been well understood for half a century. Motivated by
applications in a variety of fields ranging from mathematical physics
to stochastic networks, a theory for obliquely reflected diffusions in
piecewise smooth domains has also been developed over the last two
decades. However, in domains with rough boundaries, even the
definition of obliquely reflected diffusions is a challenge. We
discuss an approach to constructing obliquely reflected Brownian
motions (ORBMs) in a large class of bounded, simply connected planar
domains that, as a by-product, also provides a new characterization of
ORBMs in bounded smooth planar domains. The class of processes we
construct also includes certain processes with jumps like
excursion-reflected Brownian motions, which have arisen in the study
of SLE. This talk is based on works with Chris Burdzy, Zhenqing Chen
and Donald Marshall.
May 4, 3 to 4pm
Robin Pemantle (University of Pennsylvania)
Three problems in discrete probability
I will talk for roughly 5 minutes on each of three topics,
after which the audience will decide which topic
should occupy the remaining half+ hour.
Topic #1: start with a Poisson process of particles on
the line; evolve by a rule in which large intervals
eat smaller intervals; what happens?
Topic #2: Make a random set M containing each positive integer
n independent with probability 1/n. Let S be the set of sums
of elements of M. How many independent copies of S must be
intersected to arrive at a finite set, and what does this
question have to do with computational Galois theory?
Topic #3: Construct a random entire function, f, whose
zeros form a unit intensity Poisson process on the real
line. Differentiate f repeatedly, and let Ln
be the joint law of the zeros of the nth derivative of
f. What is the limit as n → infinity of Ln,
Ioana Dumitriu (University of Washington)
Two clustering problems involving the Stochastic Block Model
Clustering and community detection in large networks are important
problem with a wide spectrum of applications, from marketing to
machine learning and from genomics to social sciences. Clustering, the
non-overlapping case, has been studied for decades in settings
increasingly general; one of the most widely-used models is the
Stochastic Block Model (SBM), where the set of vertices is partitioned
in a number of subsets, and an independent Erdos-Renyi graph is
constructed on each of these sets; the inter-set edges are then given
by an independent multipartite Erdos-Renyi graph. Given the adjacency
matrix of the large graph, the problem consists of devising an
algorithm that identifies (correctly or approximately) the original
vertex partition (or showing no such algorithm exists).
Even when making several simplifications, the problem is difficult enough that the only case that has been completely solved (in the sense that all parameter reginmes have been identified) is the two equal-sized set case (binary, balanced SBM) through a concerted effort by Mossel, Neeman and Sly, parallelled by Massoulie. Inspired by their work, we have considered and analyzed a regular binary SBM, where the Erdos-Renyi models are replaced by uniform regular graphs. The first part of my talk will focus on this work, joint with Gerandy Brito, Shirshendu Ganguly, Christopher Hoffman, and Linh Tran.
The second part of my talk will involve recent work on a general SBM model involving recovery regimes (sets of parameters for which correct identification of the partition is possible). This work is joint with Maryam Fazel, Roy Han, and Amin Jalali.
Claudio Landim (IMPA)
Macroscopic fluctuation theory
Non-equilibrium stationary states describe steady flows through
macroscopic systems. Although they represent the simplest
generalization of equilibrium states, they exhibit a variety of new
phenomena. Within a statistical mechanics approach, these states have
been the subject of several theoretical investigations, both analytic
and numerical. The macroscopic fluctuation theory, based on a formula
for the probability of joint space-time fluctuations of thermodynamic
variables and currents, provides a unified macroscopic treatment of
such states for driven diffusive systems.
Conferences and Lecture Series