Columbia Probability Seminar
This page is an archive. See the current seminar page here.
The seminar covers a wide range of topics in pure and applied probability. The seminar is organized jointly by the Mathematics and Statistics departments and is run by Guillaume Barraquand, Ivan Corwin, Julien Dubedat, Ioannis Karatzas, Jeffrey Kuan, Marcel Nutz, Philip Protter, and Hao Shen .
Email the organizers at probability_seminar  at  math.columbia.edu.
The seminar usually takes place in the Mathematics Department (Math 520), on Fridays at 12 noon  1 p.m.. Directions to the Mathematics Department.
Join the Columbia Probability Seminar Mailing List.
Here is a list of upcoming probability conferences and meetings.
Spring Semester 2016
Friday, January 29, 2016

Stéphane Benoist (Columbia University)

Near critical spanning forests

Abstract: We study random spanning forests in the plane, which are slight perturbations of a uniformly chosen spanning tree (UST). We show how to relate this scaling limit to the stationary distribution of a natural Markov process on a state space of abstract graphs with edgeweights.
In the renormalization picture, the scaling limit of these spanning forests correspond to a repulsive direction around the UST fixed point.
This is a joint work with Laure Dumaz and Wendelin Werner.
Friday, February 5, 2016

Columbia—Courant Probability Day

Time: 9:00am, Feb. 5th, Math 417 Columbia University

9:009:30, Coffee, tea and light breakfast

9:3010:30, Ajay Chandra, An analytic BPHZ theorem for Regularity Structures
Abstract: When trying to tame divergences using counterterms within regularity structures there are two key things one has to verify:
(i) the insertion of the counterterm corresponds to a renormalization of the equation and is allowed by the algebraic structure of regularity structures,
(ii) there is a way to choose the value of counterterms which yield the right stochastic estimates.
This verification is difficult when the divergences become numerous and are nested/overlapping.
Recent work by Bruned, Hairer, and Zambotti provides a robust framework to systematically handle the first issue, I will describe how this can be combined with multiscale techniques from constructive field theory in order to handle the second.This is joint work with Martin Hairer.

10:4511:45, Hendrik Weber, Global wellposedness for the dynamic Phi^4_3 model the torus
Abstract: The theory of nonlinear stochastic PDEs has recently witnessed an enormous breakthrough when Hairer and Gubinelli devised methods to give an interpretation and show local wellposedness for a
class of very singular SPDEs from Mathematical Physics.
In this talk I will discuss how to extend their method to get global bounds in a prominent example, the dynamic
Phi^4 model. I will first show how to use a simple PDE argument to show global in time wellposedness for the
dynamic Phi^4 equation on the twodimensional plane. The emphasis of the talk will be on an extension of this
method which yields global in time solutions for the three dimensional Phi^4 model on the torus.
This is joint work with JeanChristophe Mourrat.

12:001:00, Weijun Xu, Large scale behaviour of phase coexistence models
Abstract: The solutions to many interesting stochastic PDEs are often obtained after suitable renormalisations. These renormalisations often change the original equation by a quantity which is infinity, but they do have concrete physical meanings. We will explain the meaning of the infinities in the context of the Phi^4_3 equation. As a consequence, we will see how this equation, interpreted after suitable renormalisations, arises naturally as the universal limit for symmetric phase coexistence models. We will also see how this universality can be lost when asymmetry is present. Based on joint works with Martin Hairer and Hao Shen.
Friday, February 19, 2016

Dirk Erhard (University of Warwick)

On a scaling limit of the parabolic Anderson model with exclusion interaction.

Abstract:
This talk is about the parabolic Anderson equation ∂u(x,t)/∂t = Δu(x,t) + [ξ(x,t)ρ]u(x,t), x∈\Z^d, t≥ 0. Here, Δ is the discrete Laplacian and the ξfield is a stationary and ergodic dynamic random environment with mean ρ that drives the equation.
The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate 2d, split into two at rate ξ ∨ 0, and die at rate (−ξ) ∨ 0.
I will focus on the case where ξ is given in terms of a simple symmetric exclusion process, i.e., ξ can be described by a field of simple random walks that move independently from each other subject to the rule that no two random walkers are allowed to occupy the same site at the same time.
I will discuss the behaviour of the equation when time and space are suitably scaled by some parameter N that tends to infinity. It turns out that in dimension two and three a renormalisation has to be carried out in order to see a nontrivial limit.
This is joint work in progress with Martin Hairer.
 Slides
Friday, March 4, 2016

Jeffrey Kuan (Columbia University)

Stochastic duality of twocomponent ASEP via symmetry of quantum groups of rank two.

Abstract:
We study two generalizations of the asymmetric simple exclusion process (ASEP) with two types of particles. Particles of type 1 can jump over particles of type 2, while particles of type 2 can only influence the jump rates of particles of type 1. We prove that the processes are selfdual and explicitly write the duality function. The construction and proofs of duality are accomplished using symmetry of the quantum groups Uq(gl_3) and Uq(sp_4), generalizing the U_q(sl_2) symmetry of ASEP.
Friday, March 11, 2016

Cameron Bruggeman (Columbia University)

Dynamics of Large RankBased Systems of Interacting Diffusions

Abstract:
We study a system of n particles evolving on the real line, whose drift and dispersion coefficients depend only on the relative rank of the particles amongst the entire system. We show that as the number of particles tends to infinity, the empirical distribution of the system converges to a deterministic limit. Using this, we can find estimates of the amount of time it takes for a particle to move from one rank to another.
March 21—23, 2016

Minerva Lectures by Scott Sheffield

Monday, March 21, 5:30–7pm, Math 507

Tuesday, March 22, 2:40–3:55pm, Math 417

Wednesday, March 23, 4:30—6pm, Math 520
Friday, April 1, 2016

Ioan Manolescu (University of Fribourg)

Planar lattices do not recover from forest fires

Abstract:
Selfdestructive percolation with parameters p, δ is obtained by taking a site percolation configuration with parameter p, closing all sites belonging to the infinite cluster, then opening every site with probability δ, independently of the rest. Call θ(p,δ) the probability that the origin is in an infinite cluster in the configuration thus obtained. For two dimensional lattices, we show the existence of δ > 0 such that, for any p > p_{c} , θ(p,δ) = 0.
This proves a conjecture of van den Berg and Brouwer, who introduced the model. Our results also imply the nonexistence of the infinite parameter forestfire model on planar lattices.
Thursday, April 7, 2016, Special Event: 1:102:10pm, Mudd 303

David Gamarnik (Sloan School of Management, Massachusetts Institute of Technology)

Finding a Large Submatrix of a Random Matrix, and the Overlap Gap Property

Abstract:
Many optimization type problems arising in random combinatorial structures and high dimensional statistics exhibit an apparent gap between the optimal values of the objective function and the values achievable by tractable algorithms. Examples include the problem of proper coloring of a random graph, finding a largest independent set of a graph, finding a sparse Principal Component of a sample covariance matrix, and many others. In our talk we consider a new example of such a gap for the problem of finding a submatrix which achieves the largest average value in a given random matrix. We will consider some known and a new algorithm for this problem, all of which produce a matrix with average value constant factor away from the globally optimal one. We then consider the overlap structure of pairs of matrices achieving a certain average value, and show that it undergoes a certain connectivity phase transition just above the value achievable by the best known algorithm.
We conjecture that the onset of this overlap gap property marks the onset of the algorithmic hardness for this problem and in fact we conjecture that this is the case for most randomly generated optimization problems.
Joint work with Quan Li (MIT)
Friday, April 8, 2016
Friday, April 15, 2016

Xin Sun (MIT)

Vertex models on random planar maps and Liouville quantum gravity

Abstract:
We start by constructing spanning trees from two classical vertex models on Euclidean lattice: 6vertex model and 20vertex model. The construction can be thought of as a generalization of Temperley bijection between dimer model and uniform spanning trees. It is closely related to Gaussian free field and SLE. Although little is known in the Euclidean case beyond conjectures, we demonstrate that the analog models on random planar maps converge to an independent coupling of imaginary geometry and Liouville quantum gravity in an unconventional but rather strong topology. This line of research is started by KenyonMillerSheffieldWilson (2015) and is further developed in a joint work with E. Gwynne and N. Holden and some ongoing projects jointly with Y. Li and S. Watson.
Wednesday, April 20, 2016, Special Event: (4:15PM Tea; 5:00PM Lecture in the Gerard D. Fischbach Auditorium; Simons Foundation 160 Fifth Ave at 21st Street, 2nd Floor).
Registration is required here.

Alexei Borodin (Massachusetts Institute of Technology)

Integrability and Universality in Probability

Abstract:
Integrable probabilistic systems are very special  they possess additional structures that make them amenable to a detailed analysis. The universality principle states that probabilistic systems from the same 'universality class' share many features. Thus, generic systems must be similar to the integrable ones in the class. In this lecture, Alexei Borodin will illustrate how these two concepts work together in examples from random matrices to random interface growth.
Friday, April 22, 2016

Roland Bauerschmidt (Harvard)

Rigidity of onecomponent plasma in 2D

Abstract:
The onecomponent plasma (OCP) is a Coulomb gas of N equal negatively charged
particles (in the continuum) confined by a potential. In two dimensions, for a
special temperature, it is integrable as a determinantal point process, and the
system can be understood is much detail. However, for generic temperatures, our
understanding is rather limited, and even basic properties of its behaviour are
not even understood heuristically. I will discuss a proof that fluctuations are much
smaller than for a Poisson process, at all finite temperatures.
This is joint work with Bourgade, Nikula, and Yau.
Friday, April 29, 2016

Jack Hanson (City College of New York)

Bigeodesics in 2d firstpassage percolation

Abstract:
Firstpassage percolation is a model for a random metric space, produced by assigning i.i.d. nonnegative weights (t_{e}) to edges of Z^{2} and considering the weighted graph metric. A number of longstanding conjectures exist regarding the behavior of infinite geodesics (infinite paths whose finite subsegments are pointtopoint geodesics). Notable among them is the claim that, under mild assumptions on the distribution of t_{e}, there should a.s. be no doubly infinite geodesic (""bigeodesic""). In the '90s, Licea and Newman showed that, under an unproven curvature assumption on the model's ""limiting shape"" (which describes the shape of large balls in the random metric), every infinite geodesic a.s. has asymptotic direction, and there is a fullmeasure set D of [0, 2π) such that for any θ_{1}, θ_{2} in D, there is no bigeodesic with ends directed in directions θ_{1} and θ_{2}. We will discuss new results on the bigeodesic conjecture showing, under the assumption that the limiting shape's boundary is differentiable, there is a.s. no bigeodesic with one end directed in any deterministic direction. This rules out existence of ground state pairs whose interface has a deterministic direction in the related disordered ferromagnet model.
May 4–6 2016, Special Event: Networks, Random Graphs and Statistics, Columbia University Department of Statistics. Website here.
Fall Semester 2015
Friday, September 18, 2015

Sumit Mukherjee (Columbia)

Limit Theorems in Graph Coloring Problems

Abstract:
In this talk the limiting distribution of the number of monochromatic edges in a uniform random coloring of any sequence of graphs will be characterized. The origin of these problems can be traced back to the classical birthday paradox, and are often helpful in the study of coincidences. I will show that the number of monochromatic edges converge in distribution to either a Poisson or a Normal when the number of colors grow to infinity. The results are universal depending solely on the limiting behavior of the ratio of the number of edges in the graph and the number of colors, and work for any graph sequence, deterministic or random. The necessary and sufficient condition for asymptotic normality when the number of colors is fixed, will also be discussed. Finally, using results from the emerging theory of graph limits, the asymptotic distribution is also characterized for any converging sequence of dense graphs. The proofs are based on moment calculations which relates to results of Erdos and Alon extremal subgraph counts.
This is based on joint work with Bhaswar Bhattacharya and Persi Diaconis.
Friday, September 25, 2015

Li—Cheng Tsai (Stanford) [11 a.m.  noon]

KPZ equation limit of interacting particle systems

Abstract:
In this talk we survey the weak universality (i.e. convergence to the KardarParisiZhang (KPZ) equation) of interacting particle systems in the KPZ universality class. Such universality is considerable more accessible when a HopfCole (HC) transformation linearizing the particle system exists. Here we derive such a HC transformation for the Higher Spin Exclusion Process (HSEP), introduced by Corwin and Petrov (2015), and hence for all known integrable models of the KPZ class in 1+1 dimensions. We further leverage this result into the weak universality of the HSEP. Next, we consider nonnearest neighbor exclusion processes, where an exact HC transformation is unavailable. We show that, for a subclass of models permitting certain gradient type conditions, an approximated form of the HC transformation exists, and prove the weak universality of the processes.
This talk is based on joint work with Ivan Corwin and joint work with Amir Dembo.

Amir Dembo (Stanford) [noon  1 p.m.]
Walking within growing domains: recurrence versus transience

Abstract:
When is simple random walk on growing in time ddimensional domains recurrent?
For domain growth which is independent of the walk, we review recent progress
and related universality conjectures about a sharp recurrence versus transience
criterion in terms of the growth rate. We compare this with the question
of recurrence/transience for time varying conductance models, where
Gaussian heat kernel estimates and evolving sets play an important role.
We also briefly contrast such expected universality with examples
of the rich behavior encountered when monotone interaction enforces the
growth as a result of visits by the walk to the current domain's boundary.
This talk is based on joint works with Ruojun Huang,
Ben Morris, Yuval Peres and Vladas Sidoravicius.
Friday, October 2, 2015


Monday, September 28, 45:30pm, 507 Math

Tuesday, September 29, 67:30pm, 507 Math

Wednesday, September 30, 4:306pm, *520* Math

Thursday, October 1, 5:307pm, 507 Math

Friday, October 2, 11:30am1pm, *520* Math
Friday, October 9, 2015, Courant Institute


Time: 9:30 am, Friday October 9th, Warren Weaver Hall, Room 512 (at Courant)

9:3010:30 Hoi Nguyen, Anti concentration of random walks and eigenvalue repulsion of random matrices.
Abstract: I will survey recent characterization results on random walks (in both abelian and nonabelian groups) which sticks to a small region unusually long. As an application, we demonstrate a Wegnertype estimate for the number of eigenvalues inside an extremely small interval for Wigner matrices of discrete type.

10:3011 Coffee break

1112 Nina Snaith, Combining random matrix theory and number theory.
Abstract: Many years have passed since the initial suggestion by Montgomery (1973) that in an appropriate asymptotic limit the zeros of the Riemann zeta function behave statistically like eigenvalues of random matrices, and the subsequent proposal of Katz and Sarnak (1999) that the same is true of families of more general Lfunctions. While this limiting behaviour is very informative, even more interesting are the intricacies involved in the approach to this limit, the understanding of which allows us to use random matrix theory in novel ways to shed light on major open questions in number theory.

121 Ramon van Handel, The norm of structured random matrices.
Abstract: Understanding the spectral norm of random matrices is a problem of basic interest in several areas of pure and applied mathematics. While the spectral norm of classical random matrix models is well understood, existing methods almost always fail to be sharp in the presence of nontrivial structure. In this talk, I will discuss new bounds on the norm of random matrices with independent entries that are sharp under mild conditions. These bounds shed significant light on the nature of the problem, and make it possible to easily address otherwise nontrivial phenomena such as the phase transition of the spectral edge of random band matrices. I will also discuss some conjectures whose resolution would complete our understanding of the underlying probabilistic mechanisms.
Friday, October 16, 2015


Tuesday, October 13th, 4:10—5:25pm, 507 Math

Friday, October 16th, 10:3011:45am, 507 Math

Tuesday, October 20th, 4:10—5:25pm, 507 Math

Thursday, October 22nd, 4:10—5:25pm, 903 SSW
Friday, October 23, 2015

Julien Reygner (Ecole des Ponts) [11 a.m.  noon]
Probabilistic interpretation of conservation laws and optimal transport in one dimension

Abstract:
We consider partial differential equations that describe the conservation of one or several quantities, possibly taking an additional dissipation mechanism into account, set on the real line. Such models are for instance relevant in gas dynamics or in the study of road traffic. When the initial data of these conservation laws are monotonic and bounded, a probabilistic theory can be developed by interpreting the solutions as cumulative distribution functions on the line. The study of the associated stochastic processes and their approximations by interacting particle systems provides a Lagrangian description of the solution, that will be used to derive existence results together with numerical schemes, as well as estimates of stability and convergence to equilibrium or traveling waves.

Pierre Le Doussal (Ecole Normale supérieure and CNRS) [noon  1pm]
From replica bethe ansatz solution of KPZ growth to noncrossing directed polymers
Friday, October 30, 2015

Yan Fyodorov (Queen Mary University of London)
A Nonlinear Analogue of May—Wigner Instability Transition

Abstract:
We study a system of N≫1 degrees of freedom individually relaxing with a rate μ and coupled via a smooth stationary random Gaussian vector field with both gradient and divergencefree components. We show that generically with increasing the ratio of the coupling strength to the individual relaxation rate the system experiences an abrupt transition from a topologically trivial phase portrait with a single stable equilibrium into topologically nontrivial regime characterized by exponential in N number of equilibria, vast majority of which is expected to be unstable. This picture is suggested to provide a nonlinear analogue of MayWigner instability transition. The analysis invokes statistical properties of the elliptic ensemble of real asymmetric matrices, and raises interesting questions about real eigenvalues of such matrices. The presentation will be based on joint work with Boris Khoruzhenko arXiv:1509.05737.
November 211, 2015


Monday, November 2nd, 4—5:30pm, 507 Math

Wednesday, November 4th, 5:307pm, 507 Math

Monday, November 9th, 4—5:30pm, 507 Math

Wednesday, November 11th, 5:307pm, 507 Math
Friday, November 6, 2015

Timo Seppalainen (University of WisconsinMadison)
Large deviations for random paths in a random medium

Abstract:
We consider the asymptotics of the free energy of a class of models of random paths in a random medium. Special cases include the muchstudied directed polymer in a random medium and random walk in random environment (RWRE). We prove the existence of the limiting free energy and obtain along the way two variational formulas for the limit. One formula minimizes over generalized gradients called cocycles, while the other one involves entropy and maximizes over invariant probability measures. As a corollary we get a fairly general quenched large deviation principle for RWRE. The behavior of these variational formulas is partly understood in some special cases. This is joint work with Firas RassoulAgha and Atilla Y{\i}lmaz.
Friday, November 13, 2015

Alexander Moll (MIT)
Random partitions and the quantum BenjaminOno hierarchy

Abstract:
Jack measures on partitions occur naturally in the study of
continuum circular loggases in generic background potentials V at
arbitrary values \beta of Dyson’s inverse temperature. Our main result
is a law of large numbers (LLN) and central limit theorem (CLT) for Jack
measures in the macroscopic scaling limit, which corresponds to the
large N limit in the loggas. Precisely, the emergent limit shape and
macroscopic fluctuations of profiles of these random Young diagrams are
the pushforwards along V of the uniform measure on the circle (LLN) and
of the restriction to the circle of a Gaussian free field on the upper
halfplane (CLT), respectively. At the critical inverse temperature
\beta=2, this recovers Okounkov’s LLN for Schur measures (2003) and
coincides with BreuerDuits’ CLT for biorthogonal ensembles (2013).
Our limit theorems follow from an allorder expansion (AOE) of joint
cumulants of linear statistics, which has the same form as the allorder
1/N refined topological expansion for the loggas on the line due to
ChekhovEynard (2006) and BorotGuionnet (2012). To prove our AOE, we
rely on the Lax operator for the quantum BenjaminOno hierarchy with
periodic profile V exhibited in collective field variables by
NazarovSklyanin (2013). The characterization of the limit laws as
pushforwards follows from factorization formulas for resolvents of
Toeplitz operators with symbol V due to Krein and CalderónSpitzerWidom
(1958).
Thursday, November 19, and Friday, November 20, 2015


The Fourteenth Northeast Probability Seminar (NEPS) will be held in Room 912/914 of the Kimmel Center at NYU on November 19th and Room 109 of the Courant Institute, NYU on November 20th, 2015.

* Jian Ding (University of Chicago)
"Some geometric aspects for twodimensional Gaussian free fields."

* Francis Comets (Université Paris Diderot)
"Localization in onedimensional loggamma polymers"

* JeanFrançois Le Gall (University ParisSud Orsay)
"Firstpassage percolation on random planar graphs."

* Mykhaylo Shkolnikov (Princeton University)
"Edge of beta ensembles and the stochastic Airy semigroup."
Friday, November 27, 2015
Friday, December 4, 2015

Vu—Lan Nguyen (Universite Paris—Diderot)
Variants of geometric RSK, geometric PNG and the multipoint distribution of the loggamma polymer

Abstract:
The geometric RobinsonSchenstedKnuth (gRSK) correspondence has played an important role in the recent analysis of the directed polymer model with a logGamma potential. In particular, it has led to the confirmation of the TracyWidom GUE asymptotic distribution of its pointtopoint partition function. In a joint work with Nikos Zygouras, we reformulate the gRSK correspondence in a way that leads to explicit integral formulae for the joint distribution of the pointtopoint partition functions at different spacetime points.
Friday, December 11, 2015

Leonid Koralov (University of Maryland)
Averaging, homogenization, and large deviation results for the study of randomly perturbed dynamical systems

Abstract:
In this talk we'll discuss several asymptotic problems that can be formulated in terms of PDEs and solved using probabilistic methods. The first set of problems concerns the asymptotic behavior of solutions to quasilinear parabolic equations with a small parameter at the second order term. Here we employ an extension of the largedeviation theory of Freidlin and Wentzell. Another set of problems concerns equations with a small diffusion term, where the firstorder term corresponds to an incompressible flow, possibly with a complicated structure of flow lines. Here we use an extension of the averaging principle. Finally, we'll consider equations with a small diffusion term with periodic coefficients in a large domain. Depending on the relation between the parameters, either averaging or homogenization need to be applied in order to describe the behavior of solutions. We'll discuss the transition regime. Different parts of the talk are based on joint results with D.Dolgopyat, M. Freidlin, M. Hairer, and Z. PajorGuylai.
Spring Semester 2015
Friday, January 30, 2015

LouisPierre Arguin (CUNY / University of Montreal)
Maxima of logcorrelated Gaussian fields and of the Riemann Zeta function on the critical line

Abstract:
A recent conjecture of Fyodorov, Hiary & Keating states that the
maxima of the Riemann Zeta function on a bounded interval of the
critical line behave similarly to the maxima of a specific class of
Gaussian fields, the socalled logcorrelated Gaussian fields. These
include important examples such as branching Brownian motion and the 2D
Gaussian free field. In this talk, we will highlight the connections
between the number theory problem and the probabilistic models. We will
outline the proof of the conjecture in the case of a randomized model
of the Zeta function. We will discuss possible approaches to the
problem for the function itself. This is joint work with D. Belius
(NYU) and A. Harper (Cambridge).
Friday, February 6, 2015
Fastslow systems with chaotic noise.

Abstract:
It has long been observed that multiscale systems, particularly those
in climatology, exhibit behavior typical of stochastic models, most
notably in the unpredictability and statistical variability of events.
This is often in spite of the fact that the underlying physical model
is completely deterministic. One possible explanation for this
stochastic behavior is deterministic chaotic effects. In fact, it has
been well established that the statistical properties of chaotic
systems can be well approximated by stochastic differential equations.
In this talk, we focus on fastslow ODEs, where the fast, chaotic
variables are fed into the slow variables to yield a diffusion
approximation. In particular we focus on the case where the chaotic
noise is multidimensional and multiplicative. The tools from rough path
theory prove useful in this difficult setting.
Friday, February 13, 2015

Riddhipratim Basu (UC Berkeley)
Maximal Increasing Subsequence with a Defect Line and Lebowitz's Slow Bond Problem

Abstract:
For a Poisson process in the plane with intensity 1, the distribution
of the maximum number of points on an oriented path from (0,0) to (N,N)
has been studied in detail, culminating in BaikDeiftJohansson's
celebrated TracyWidom fluctuation result. We consider a variant of the
model where one adds, on the diagonal, some additional points according
to an independent one dimensional Poisson process with rate \lambda.
The question of interest here is whether for all positive values of
\lambda, this results in a change in the law of large numbers for the
the number of points on a maximal path. A closely related question
comes from a variant of Totally Asymmetric Simple Exclusion Process,
introduced by Janowsky and Lebowitz. Consider a TASEP in 1dimension,
where the bond at the origin rings at a slower rate r<1. The
question is whether for all values of r<1, the single slow bond
produces a macroscopic change in the system. We provide affirmative
answers to both the questions. Based on joint work with Vladas
Sidoravicius and Allan Sly.
Friday, February 20, 2015

Johannes Ruf (University College London)
Convergence of local supermartingales and Novikovtype conditions for processes with jumps

Abstract:
In the first part of the talk, we characterize the event of convergence
of a local supermartingale. Conditions are given in terms of its
predictable characteristics and jump measure. Furthermore, it is shown
that L^1boundedness of a related process is necessary and sufficient
for convergence. The notion of extended local integrability plays a key
role. In the second part of the talk, we provide a novel proof for the
sufficiency of NovikovKazamaki type conditions for the martingale
property of nonnegative local martingales with jumps. The proof is
based on explosion criteria for related processes under a possibly
nonequivalent measure. This is joint work with Martin Larsson.
Friday, February 27, 2015, Columbia Mathematics Department, room 520, special starting time: 9:30am (breakfast at 9:00am in room 508)

Columbia / Courant Joint Probability Seminar Series:

Speakers: Vadim Gorin (MIT), Richard Kenyon (Brown), Greta Panova (U Penn)
More information here.
Friday, March 6, 2015

Columbia Princeton Probability Day (Princeton)

Speakers: Davar Khoshnevisan (Utah), Fraydoun Rezakhanlou (Berkeley),
Prasad Tetali (Georgia Tech), Balint Virag (Toronto), Alex Drewitz
(Columbia), Leonid Petrov (Virginia)
More information here.
March 911, 2015
Friday, March 27, 2015
High temperature limits for (1+1)d directed polymer with heavytailed disorder.

Abstract:
The directed polymer model at intermediate disorder regime was
introduced by AlbertsKhaninQuastel (2012). It was proved that at
inverse temperature beta n^{gamma} with gamma=1/4 the partition
function, centered appropriately, converges in distribution and the
limit is given in terms of the solution of the stochastic heat
equation. This result was obtained under the assumption that the
disorder variables posses exponential moments, but its universality was
also conjectured under the assumption of six moments. We show that this
conjecture is valid and we further extend it by exhibiting classes of
different universal limiting behaviors in the case of less than six
moments. We also explain the behavior of the scaling exponent for the
logpartition function under different moment assumptions and values of
gamma. Based on joint work with Nikos Zygouras.
Friday, April 3, 2015
A two scale proof of the EyringKramers formula

Abstract:
We consider a driftdiffusion process on a smooth potential landscape
with small noise. We give a new proof of the EyringKramers formula
which asymptotically characterizes the spectral gap of the generator of
the diffusion. The proof is based on a refinement of the twoscale
approach introduced by Grunewald, Otto, Villani, and Westdickenberg and
of the meandifference estimate introduced by Chafai and Malrieu. The
new proof exploits the idea that the process has two natural
timescales: a fast timescale resulting from the fast convergence to a
metastable state, and a slow timescale resulting from exponentially
long waiting times of jumps between metastable states. A nice feature
of the argument is that it can be used to deduce an asymptotic formula
for the logSobolev constant, which was previously unknown (joint work
with Andre Schlichting).
April 810, 2015
Friday, April 10, 2015

Wendelin Werner (ETH Zurich) [11 a.m. — noon]
A simple graphvalued Markov processes and renormalization

Abstract:
We
describe a simple graphvalued Markov process and will explain how it
can provide an elementary approach to some renormalization group ideas
and questions. In particular, a (small) perturbation of the conjectured
continuous scaling limits of critical models appear as stationary
distributions for these Markov chains. We also point out how to relate
this approach to some of the known results concerning scaling limits
for some two dimensional models (the relation to uniform spanning tree
is joint work with Stéphane Benoist and Laure Dumaz).

Boguslaw Zegarlinski (Imperial College London) [12 noon — 1 p.m.]
Ergodicity and smoothing of Markov semigroups with Hoermander type generators in infinite dimensions

Abstract:
This will be about applications of techniques based on
generalised gradient bounds
Friday, April 17, 2015
Scaling limits for a random conductance problem

Abstract:
This talk is about a random conductance model. Imagine an electrical network defined on the integer lattice Z^d with random, iid conductances assigned to the links between nearest neighbors. The voltage at each node and the current across each link is determined by solving a linear system of equations which is analogous to solving a divergenceform elliptic PDE with random coefficients. An important object of study is known as the ``corrector", which describes the random fluctuations in voltage that result from a macroscopic voltage gradient across the network. I'll explain how, in dimension 3 or more, a rescaled version of this discrete random object converges to a Gaussian field with covariance structure that is similar in some ways to a Gaussian free field. I'll also describe a central limit theorem for the effective conductivity of a finite network as the size of the network grows. This includes joint work with JeanChristophe Mourrat, and with Antoine Gloria.
April 2024, 2015
Friday, April 24, 2015
A class of (2+1)dimensional growth process with explicit stationary measure
Abstract:
We introduce a class of (2 +
1)dimensional random growth processes, that can be seen as asymmetric
random dynamics of discrete interfaces. Interface configurations
correspond to height functions of dimer coverings of the infinite
hexagonal or square lattice. “Asymmetric” means that the interface has
an average nonzero drift. When the asymmetry parameter p − 1/2 equals
zero, the infinitevolume Gibbs measures pi_\rho (with given slope
\rho) are stationary and reversible. When p\neq 1/2, \pi_\rho is not
reversible any more but, remarkably, it is still stationary. In such
stationary states, one finds that the height function at a given point
x grows linearly with time t with a nonzero speed, := <(hx(t) −
hx(0))> = v t and that the typical fluctuations of Q_x(t) are
smaller than any power of t.
For the specific case p = 1 and in the case of the hexagonal lattice,
the dynamics coincides with the “anisotropic KPZ growth model” studied
by A. Borodin and P. L. Ferrari. For a suitably chosen initial
condition (that is very different from the stationary state), they were
able to determine the hydrodynamic limit and the interface
fluctuations, exploiting the fact that some spacetime correlations can
be computed exactly, and predicted stationarity of Gibbs measures.
Friday, May 1, 2015

Hugo DuminilCopin (Geneva)
A new proof of exponential decay of correlations in subcritical percolation and Ising models

Abstract:
We provide a new proof of exponential decay of correlations for subcritical Bernoulli percolation on Z^d. The proof is based on an alternative definition of the critical point. The proof extends to the Ising model and to infiniterange models on infinite locally finite transitive graphs. It also provides a meanfield lower bound for the explosion of the infinitecluster density in the supercritical regime.
Joint work with Vincent Tassion.
Wednesday, May 13, 2015

Claudio Landim (IMPA Rio de Janeiro) [12 noon — 1 p.m., room: Math 507]
Condensing zerorange processes

Abstract:
Zerorange processes with decreasing jump rates exhibit a condensation
transition, where a positive fraction of all particles condenses on a
single lattice site when the total density exceeds a critical
value.
Consider a condensing zerorange process with $N$ particles evolving
on a fixed and finite set $S$. In the diffusive time scale $N^2$ the
fraction of particles at each site evolves as a diffusion whose drift
is unbounded. The process remains absorbed at the boundary once it
attains it, and it performs, after this hitting time, a diffusion on a
lower dimensional simplex, similar to the original one, until all
particles concentrate on a single site.
In a longer time scale $N^{1+\alpha}$, $\alpha>1$, the site which
concentrates all particles evolves as a random walk on $S$ whose
transition rates are proportional to the capacities of the underlying
random walk.
Fall Semester 2014
Friday, September 12, 2014
The subleading order of two dimensional cover times

Abstract:
The epsiloncover time of the two dimensional torus by Brownian motion
is the time it takes for the process to come within distance
epsilon>0 from any point. Its leading order in the small
epsilonregime has been established by Dembo, Peres, Rosen and Zeitouni
[Ann. of Math., 160 (2004)]. In this talk I will present a recent
result identifying the second order correction. This correction term
arises in an interesting way from strong correlations in the field of
occupation times, and in particular from an approximate hierarchical
structure in this field. Our method draws on ideas from the study of
the extremes of branching Brownian motion. Joint work with Nicola
Kistler.
Friday, September 19, 2014

Roland Bauerschmidt (Harvard)
Selfavoiding walks and spins in 4 dimensions

Abstract:
The weakly selfavoiding walk is a model of random polymers predicted
to show the same universal behavior as the strictly selfavoiding walk.
Similarly, the ncomponent \varphi^4 model is a ferromagnetic spin
model predicted to have the same universal behavior as the Ising model
for n=1, the rotor model for n=2, and the classical Heisenberg model
for n=3. It is known that these apparently different models are closely
related in the sense that selfavoiding walks can be seen as spins with
"n=0" components. I will present results for both models in 4
dimensions. In particular, we show that the susceptibility has a
logarithmic correction to the meanfield scaling behavior with exponent
(n+2)/(n+8) for the logarithm, thus 1/4 for the weakly selfavoiding
walk. For the specific heat, we show a similar fractional logarithmic
divergence for 0<4,
a double logarithmic divergence for n=4, and that it is bounded for
n>4 components. This is joint work with David Brydges and Gordon
Slade. The proofs rely on recently developed methods for
renormalisation group analysis.
Friday, September 26, 2014

Omer Angel (UBC Vancouver)
Unimodular hyperbolic planar graphs

Abstract:
We study random hyperbolic planar graphs by using their circle packing
embedding to connect their geometry to that of the hyperbolic plane.
This leads to several results: Identification of the Poisson and
geometric boundaries, a connection between hyperbolicity and a form of
nonamenability, and a new proof of the BenjaminiSchramm recurrence
result. Based on works with subsets of Martin Barlow, Ori
GurelGurevich, Tom Hutchcroft, Asaf Nachmias and Gourab Ray.
Friday, October 3, 2014
Burgers equation with random forcing

Abstract:
The Burgers equation is one of the basic nonlinear evolutionary PDEs.
The study of ergodic properties of the Burgers equation with random
forcing began in 1990's. The natural approach is based on the analysis
of optimal paths in the random landscape generated by the random force
potential. For a long time only compact cases of the Burgers dynamics
on a circle or bounded interval were understood well. In this talk I
will discuss the Burgers dynamics on the entire real line with no
compactness or periodicity assumption on the random forcing. The main
result is the description of the ergodic components and existence of a
global attracting random solution in each component. The proof is based
on ideas from the theory of first or last passage percolation. The
kicked forcing case is an extension of the Poissonian forcing case
considered in a joint work with Eric Cator and Kostya Khanin.
Friday, October 10, 2014, Courant Institute, Warren Weaver Hall, 512

Columbia / Courant Joint Probability Seminar Series:

Speakers: Yuri Bakhtin (Courant), Paul Bourgade (Courant), Ivan Corwin (Columbia) and Eyal Lubetzky (Courant)
More information here.
Friday, October 17, 2014

Andrey Sarantsev (U Washington)
Collisions of competing Brownian particles

Abstract:
Consider a finite system of Brownian particles on the real line, with
each particle having drift and diffusion coefficients depending on its
current rank relative to other particles. A triple collision occurs
when three or more particles occupy the same position at the same time.
We find a necessary and sufficient condition for absence of triple
collisions. We also investigate some other types of collisions. This
continues the work by Ichiba, Karatzas and Shkolnikov.
Friday, October 24, 2014
Friday, October 31, 2014

Guillaume Barraquand (Paris 7) [11 a.m. — 12 noon]
KPZ scaling theory for integrable exclusion processes

Abstract:
The KPZ scaling theory provides a general method to compute all
modeldependent constants arising in limit theorems for a large class
of exclusion processes. The validity of this heuristic approach is
rigorously proved only for a few exactly solvable models. In this talk,
we will discuss how the theory applies for qdeformed exclusion
processes introduced by BorodinCorwin and Povolotsky : The qTASEP and
the qHahn TASEP. We will also introduce a twosided generalization of
the qHahn TASEP that preserve the integrable structure and further
confirm KPZ scaling theory. This is a joint work with Ivan Corwin.

Michael Damron (U Indiana) [12 noon — 1 p.m.]
Rate of convergence of the mean for subadditive ergodic sequences

Abstract:
For a subadditive ergodic sequence {X_{m,n}}, Kingman's theorem gives
convergence for the terms X_{0,n}/n to some nonrandom number g. In
this talk, I will discuss the convergence rate of the mean EX_{0,n}/n
to g. This rate turns out to be related to the size of the random
fluctuations of X_{0,n}; that is, the variance of X_{0,n}, and the main
theorems I will present give a lower bound on the convergence rate in
terms of a variance exponent (if it exists). The main assumptions are
that the sequence is not diffusive (the variance does not grow
linearly) and that it has a weak dependence structure. Various
examples, including first and last passage percolation, bin packing,
and longest common subsequence fall into this class. This is joint work
with Tuca Auffinger and Jack Hanson.
Friday, November 7, 2014
A conformally invariant metric on CLE_{4}

Abstract:
We will discuss an exploration process, introduced by Wendelin Werner and Hao Wu, in which conformal loop ensemble (CLE_{κ})
loops grow uniformly from the boundary of a domain. We relate this
process in the case κ = 4 to the set of level loops of the
zeroboundary Gaussian free field, and we use this point of view to
show that the exploration process is a deterministic function of the
CLE loops. We describe how this gives rise to a conformally invariant
metric on CLE_{4}, which we conjecture can be given a natural geometric interpretation. Based on joint work with Scott Sheffield and Hao Wu.
Friday, November 14, 2014

Licheng Tsai (Stanford) [11 a.m. — 12 noon]
Infinite Dimensional Stochastic Differential Equations for Dyson’s Brownian Motion

Abstract:
Dyson's Brownian Motion (DBM) describes the evolution of the spectra of
certain random matrices, and is governed by a system of stochastic
differential equations (SDEs) with a singular, longrange interaction.
In this talk I will outline a construction of the strong solution of
the infinite dimensional SDE that corresponds to the bulk limit of DBM.
This is a pathwise construction that allows an explicit space with
generic configurations. The ideas used further lead to a proof of the
pathwise uniqueness of the solution and of the convergence of the
finite to infinite dimensional SDE.

Brian Rider (Temple) [12 noon — 1 a.m.]
Matrix Dufresne Identities

Abstract:
The classical Dufresne identity relates the distribution of the
infinite time integral of geometric Brownian motion (a quantity of
relevance in, for example, mathematical finance) to an inverse gamma
law. This fact, along with various extensions, lurks behind the scenes
in the geometric lifting of Pitman’s 2MX formula due to MatsumotoYor,
as well as in O’Connell’s recent work on the quantum Toda lattice. I
will describe what can be said thus far in a matrix setting, when the
Wishart distribution replaces the gamma, and Brownian motion on the
general linear group steps in for the onedimensional geometric
Brownian motion. This is joint work with B. Valkó (University of
Wisconsin).
Thursday, November 20, and Friday, November 21, 2014
Tuesday, December 2, 2014 (joint probability / applied math seminar)

Tom Trogdon (Courant) [2:45 p.m. — 3:45 a.m., 214 Mudd Building]
Universality in numerical computations with random data

Abstract:
How do numerical algorithms perform statistically? To begin to analyze
this broad question we follow Pfrang, Deift and Menon (2012) and
diagonalize random matrices with integrable systems. Universal
fluctuations are observed in the time to first deflation. We contrast
this empirical study with analytical results for naive eigenvalue
algorithms. We also consider the conjugate gradient, GMRES and genetic
algorithms. Universality in iteration count is observed and new
insights into random matrix theory are discussed. Finally, we see
realworld applications of these ideas in a model of neural computation
and in data analysis. This is joint work with P. Deift, G. Menon and S.
Olver.
Friday, December 5, 2014

Wolfgang König (WIAS / TU Berlin)
A variational formula for the free energy of an interacting manybody system

Abstract:
We consider N bosons in a box in the ddimensional space in a large box
under the influence of a mutually repellent pair potential. The
particle density is kept fixed and positive. Our main result is the
identification of the limiting free energy at positive, sufficiently
high temperature in terms of an explicit variational formula. The
thermodynamic equilibrium is described by the symmetrised trace of the
negative exponential of the corresponding Hamilton operator. The
wellknown FeynmanKac formula reformulates this trace in terms of N
interacting Brownian bridges. Due to the symmetrisation, the bridges
are organised in an ensemble of cycles of various lengths. The novelty
of our approach is a description in terms of a marked Poisson point
process whose marks are the cycles. This allows for an asymptotic
analysis of the system via a largedeviation analysis of the stationary
empirical field. The resulting variational formula ranges over random
shiftinvariant marked point fields and optimizes the sum of the
interaction and the relative entropy with respect to the reference
process. Our formula is not able to express the unboundedly long
cycles; as a result we derive only lower and upper bounds. Our results
and their shortcomes are at the heart of BoseEinstein condensation.
Monday, December 15, and Tuesday, December 16, 2014 (Joseph Fels Ritt Lectures)

AlainSol Sznitman (ETH Zurich) [4:30 p.m., Davis Auditorium, 412 Schapiro Center, CEPSR 530 West 120th Street (between Broadway and Amsterdam)]
Disconnection, interlacements, and the Gaussian free field

Click here for further information
Spring Semester 2014
Friday, Jan 24, 2014
Rough paths theory and Gaussian processes

Sebastian Riedel (TU Berlin)
Friday, Jan 31, 2014

Jürg Fröhlich (ETH Zürich & IAS Princeton)
Quantum Probability Theory and the Emergence of Facts in Quantum Theory

Abstract:
This talk represents an experiment in explaining quantum theory to
mathematicians. First, it is explained on a simple example in which way
quantum probability theory differs from classical probability theory. A
dictionary between the two fields is sketched. It is then outlined what
"quantum dynamical systems" are, and the key concepts of "information
loss" and "entanglement" are introduced. Subsequently, the theory of
von 'Neumann Measurements' is briefly described. The talk concludes
with the theory of 'NonDemolition Measurements' and, on some examples,
will hopefully clarify how "purification" of states arises and how
"facts" emerge in quantum mechanics.
Friday, Feb 7, 2014
Exact Simulation of Multidimensional Reflected Brownian Motion

Abstract:
A number of algorithms have been produced recently to simulate exactly
from one dimensional diffusions. These are remarkable because the
transition density of a diffusion is typically not known in closed
form. One uses Lamperti's transformation to induce a constant diffusion
coefficient. Then, a key step is to propose a sample from Brownian
motion. The likelihood ratio can often be localized and the socalled
acceptance step can be executed using a Poisson thinning procedure.
While similar strategies can be used for one dimensional reflected
diffusions, Multidimensional reflected Brownian motion (RBM) calls for
a new methodology. This is mainly because there is no simulatable
process that can be used as a proposal and the transition density of
multidimensional RBM is not known. In this talk we explain this new
methodology. (This talk is based on joint work with Chris Dolan and
Karthyek Murthy).
Friday, Feb 14, 2014

Manuel Cabezas (IMPA Rio de Janeiro) [11 a.m. — 12 noon]

Siterecurrence for twotype annihilating random walks
Abstract: We consider a particle system where particles can be of two
types, A or B. The particles perform independent random walks on $Z^d$
until they meet a particle of opposite type. At that time both
particles annihilate and are no longer present in the system. Particles
of the same type do not interact. We assume that the initial
configuration is independent and identically distributed over $x$ in
$Z^d$. We prove that the system is siterecurrent, that is, we prove
that, almost surely, the origin is visited infinitely often by
particles. (joint work with L. Rolla And V. Sidoravicius)

Georg Menz (Stanford) CANCELLED

A two scale proof of the EyringKramers formula
Abstract: We consider a diffusion on a potential landscape which is
given by a smooth Hamiltonian in the regime of small noise. We give a
new proof of the EyringKramers formula for the spectral gap of the
associated generator of the diffusion. The proof is based on a
refinement of the twoscale approach introduced by Grunewald, Otto,
Villani, and Westdickenberg and of the meandifference estimate
introduced by Chafai and Malrieu. The EyringKramers formula follows as
a simple corollary from two main ingredients : The first one shows that
the Gibbs measures restricted to a domain of attraction has a "good"
Poincaré constant mimicking the fast convergence of the diffusion to
metastable states. The second ingredient is the estimation of the
meandifference by a new weighted transportation distance. It contains
the main contribution of the spectral gap, resulting from exponential
long waiting times of jumps between metastable states of the diffusion.
This new approach also allows to derive sharp estimates on the
logSobolev constant. (joint work with A. Schlichting)
Friday, Feb 21, 2014
Vacant set of random walk on discrete torus

Abstract:
We prove a phase transition in the behaviour of certain macroscopic
observable on the vacant set of random walk on the ddimensional
discrete torus, d>2. To this end we discuss a technique of coupling
of Markov chains so that their ranges almost coincide all the time
(joint work with A. Teixeira).
Friday, Feb 28, 2014
Friday, Mar 7, 2014

Rodrigo Bañuelos (Purdue) CANCELLED
tba
Friday, Mar 14, 2014
Uniformity of the late points of random walk on \Z^d_n for d \ge 3

Perla Sousi (U Cambridge) [11 a.m. — 12 noon]

Michele Salvi (TU Munich) [12 noon — 1 p.m.]
Homogenization in the Random Conductance Model: A CLT for the Effective Conductance

Abstract: In nature, most materials have a rather complicated
microscopic structure. Nevertheless, the global properties of the
material (such as heat or electric conduction) can be described by
differential equations with smooth coefficients. An explaination to
this behavior is offered by Homogenization Theory: The rapid
oscillations of local coefficients caused by impurities average out, or
homogenize, at a macroscopic scale. The Effective Conductance of random
electrical networks, i.e. the minimum of the Dirichlet Energy, is a
prominent example of a homogenizing quantity. When scaled by the volume
of the domain, the Effective Conductance was already known in the 80s
to have an almost sure constant limit, but the order of fluctuations
has remained an open problem for almost thirty years. In our work, we
were able to derive a (nondegenerate) Central Limit Theorem for this
quantity, albeit under rather restrictive hypothesis (i.i.d.
conductances on the network with small ellipticity contrast, square
domains and linear boundary conditions). The proof is based on the
corrector method and the Martingale Central Limit Theorem, while a key
integrability condition is furnished by the Meyers estimate. (joint
work with Marek Biskup and Tilman Wolff)
Friday, Mar 21, 2014

no seminar (Spring break)
Friday, Mar 28, 2014

Frank Aurzada (TU Darmstadt)
Persistence probabilities

Abstract:
Persistence probabilities concern the question of the probability that
a stochastic process has a long excursion. For example one asks for the
rate of P( \sup_{0\leq t\leq T} X_t \leq 1 ) as T tends to infinity.
Further, one studies the behavior of the process conditioned on having
such an excursion. The talk will survey some recent results in this
area, in particular for fractionally integrated processes and for
fractional Brownian motion.
Friday, Apr 4, 2014

Soumik Pal (U Washington)
The geometry of relative arbitrage

Abstract:
Suppose we do not impose stochastic models on how stock prices will
evolve in the future. Is it possible, by active trading, to do better
than a market index (say, S&P 500)? In this second part of a
twopart talk^* (which can nevertheless be understood independently) we
will prove the following surprising fact in complete generality. If we
restrict ourselves to portfolios that are functions of the current
stock prices, there is exactly one class of trading strategies that
achieves this goal. Remarkably, these strategies are produced as
solutions of MongeKantorovich optimal transport problem on the
multidimensional unit simplex with a cost function that can be
described as the log partition function. The ideas are an interplay
between convex analysis, probability theory, and the geometry of the
unit simplex. Based on joint work with Leonard Wong.
^* See here for the first part of the talk.
Friday, Apr 11, 2014

Columbia Princeton Probability Day
Friday, Apr 18, 2014

Laurent SaloffCoste (Cornell)
Edge traffic on graphs and random walks on free solvable groups

Abstract:
What is the probability that a simple random walk on the square grid
return to its starting point at time n after traversing each edge an
equal number of time in both direction? The step 2 free solvable group
with r generators is the quotient of the free group on r generators by
its second commutator subgroup. What is the probability that the simple
random walk on this group is back to its starting point at time n? And
how are these questions related to each other? In discussing these
questions, I will avoid technical details and focus on the description
of the problems and results and some of the main ideas. (joint work
with Tianyi Zheng, Stanford)
Apr 2125, 2014
Friday, Apr 25, 2014
Asymmetric Cauchy distribution and the destruction of large random recursive trees

Abstract:
The probability mass function $1/j(j+1)$ for $j\geq 1$ belongs to the
domain of attraction of a completely asymmetric Cauchy distribution.
The purpose of the talk is to review some of applications of this
simple observation to limit theorems related to the destruction of
random recursive trees.
Specifically, a random recursive tree of size $n+1$ is a tree chosen
uniformly at random amongst the $n!$ trees on the set of vertices
$\{0,1, …, n\}$ such that the sequence of vertices along any segment
starting from the root $0$ increases. One destroys this tree by
removing its edges one after the other in a uniform random order. It
was first observed by Iksanov and M\"ohle that the central limit
theorems for the random walk with step distribution given above
explains the fluctuations of the number of cuts needed to isolate the
root. We shall discuss further results in the same vein.
Friday, May 16, 2014 (both talks take place in Mathematics, room 520)

Siva Athreya (ISI Bangalore) [11 a.m. — 12 noon]
Invariance principle for variable speed random walks on trees

Abstract:
We consider stochastic processes on $(T,r)$ a complete, locally compact
treelike metric spaces on their "natural scale" with boundedly finite
speed measure $\nu$. If $(T,r)$ is a discrete tree, then the process is
a continuous time nearest neighbor random walk which jumps at rate
depending on $\nu$. If $(T,r)$ is pathconnected, then the process has
continuous paths and equals the so called $\nu$Brownian motion on
$(T,r)$. In this talk we will discuss an invariance principle which
shows that speed$\nu_n$ motions on $(T_n,r_n)$ converge weakly in path
space to the speed$\nu$ motion on $(T,r)$ provided that the underlying
triples of metric measure spaces converge in the GromovHausdorffvague
topology. This is joint work with Anita Winter and Wolfgang Lohr.

Rongfeng Sun (NUS Singapore) [12 noon  1 p.m.]
Polynomial chaos and scaling limits of disordered systems

Abstract:
Inspired by recent work of Alberts, Khanin and Quastel, we formulate
general conditions ensuring that a sequence of multilinear polynomials
of independent random variables (called polynomial chaos expansions)
converges to a limiting random variable, given by an explicit Wiener
chaos expansion over the ddimensional white noise. A key ingredient in
our approach is a Lindeberg principle for polynomial chaos expansions,
which extends earlier work of Mossel, O'Donnell and Oleszkiewicz. These
results provide a unified framework to study the continuum and weak
disorder scaling limits of statistical mechanics systems that are
disorder relevant, including the disordered pinning model, the
longrange directed polymer model in dimension 1+1, and the
twodimensional random field Ising model. This gives a new perspective
in the study of disorder relevance, and leads to interesting new
continuum models that warrant further studies. Joint work with F.
Caravenna and N. Zygouras.
Fall Semester 2013
Friday, Sept 6, 2013
The critical curve of the Copolymer Model at weak coupling

Abstract:
We will present the Copolymer Model introduced by Sinai, used to describe a
heteregeneous polymer chain (say composed of hydrophilic or lipophilic
monomers) lying at the interface between two solvents (say water and oil).
When increasing the temperature, one observes a localization/delocalization
phase transition: at low temperature, the polymer stays close to the
interface to maximize the matches, whereas at high temperature, it wanders
in one of the two solvents.
We focus here on the critical curve, that separates the two regimes in this
phase transition. In particular, we study its weak coupling limit, which is
known to be universal. The value of the limit has however been subject to
discussions, leading to contradictory conjectures. The result we will
present is that we are actually able to compute the limit, under the
assumption that the underlying return distribution has a powerlaw tail
with finite mean.
(joint work with F. Caravenna, J. Poisat, R. Sun, N. Zygouras)
Friday, Sept 13, 2013

Ivan Corwin (Columbia, Clay Mathematics Institute, MIT)
Spectral theory for the qBoson particle system

Abstract: We develop spectral theory for the generator of the qBoson
particle system. Our central result is a Plancherel type isomorphism
theorem for this system. This theorem has various implications. It
proves the completeness of the Bethe ansatz for the qBoson generator
and consequently enables us to solve the Kolmogorov forward and
backward equations for general initial data. Owing to a Markov duality
with qTASEP, this leads to moment formulas which characterize the
fixed time distribution of qTASEP started from general initial
conditions. The theorem also implies the biorthogonality of the left
and right eigenfunctions. Some limits of our results relate to directed
polymers and the KardarParisiZhang equation.
Friday, Sept 20, 2013
Identifying and Representing SelfSimilar Processes

Abstract: The systematic study of selfsimilar Markov processes goes
back to the seminal article of John Lamperti from '72. For selfsimilar
processes with positive trajectories a relation to Lévy processes is
nowadays wellknown whereas a general theory for realvalued
selfsimilar process is still to be built. I will discuss the present
knowledge and introduce an approach via singular SDEs.
Friday, Sept 27, 2013
Laws of
the Iterated Logarithm for Selfnormalized Levy Processes at Zero
Friday, Oct 4, 2013, 10am1:30pm in room Math 520, Columbia University

Columbia / Courant Joint Probability Seminar Series:

KardarParisiZhang Universality
Speakers:
Herbert
Spohn (TU Munich), Jeremy Quastel (U. Toronto), and Leonid Petrov
(Northeastern)
More information here.
Friday, Oct 11, 2013

Omer Tamuz (Microsoft/MIT)
Product group rigidity using probability

Abstract: In this talk we will explain how, using probabilistic constructions such as
random walks on groups and their Poisson boundaries, one can study the
algebraic properties of a group, as well as the properties of the
probability spaces on which it acts.

Joint work with Yair Hartman
Friday, Oct 18, 2013

Gaetan Borot (Max Planck Institute)
Allorder asymptotics in beta ensembles in the multicut regime

Abstract: The beta ensemble is a particular model consisting of N
strongly correlated real random variables. For specific values of beta,
it is be realized by the eigenvalues of a random hermitian matrix whose
distribution is invariant by conjugation, and in this case the model is
exactly solvable in terms of orthogonal polynomials, and provide
solutions to the Toda chain equations. I will give present results on
the large N asymptotics up to O(N^{infinity}) of the partition
function, and the moments of the x_i's, away from critical points. I
will give an idea about the methods we use, which do not not rely on
techniques of exactly solvable models. Under fairly general
assumptions, the x_i accumulate in the large N limit on a collection of
(g + 1) segments of the real line. The asymptotic behavior depends much
on g: if g = 0, there is a 1/N expansion, but if g > 0, the model
features an oscillatory behavior at all orders in 1/N. As an
application, we obtain the allorder asymptotics of orthogonal
polynomials away from their zero locus, and of solutions of the Toda
chain in the continuum limit.
Based on joint work with A. Guionnet (MIT)
Friday, Oct 25, 2013

Ofer Zeitouni (Weizmann Institute/Courant)
Slowdown for branching Brownian motion

Abstract: Branching Brownian motion is closely related to solutions of the
KPPFisher equation; the distribution of the maximum is
the solution at time $t$, started from an initial step function; it lags
$c\log t$ behind a travelling wave solution. The constant $c$, which was
first computed by Bramson using probabilistic methods, is related
to the behavior of Brownian motion conditioned to remain above level $1$
up to time $t$. I will discuss a variant where the diffusivity of the
Brownian motion changes macroscopically. The correction term exhibits then
a phase transition, and in a cerrtain regime is of order $ct^{1/3}$, with
the constant $c$ related to the first zero of Airy's function. The proof involves an
obstacle problem for Brownian motion with curved boundary.
(Joint work with Pascal Maillard http://arxiv.org/abs/1307.3583)
Friday, Nov 1, 2013

Omer Angel (UBC) — canceled
Friday, Nov 8, 2013
On the conformally invariant growing mechanism in CLE4

Abstract: CLE4 is the collection of level lines of GFF. Miller and
Sheffield give the first coupling between GFF and CLE4 in the way that
loops in the CLE4 are the outmost $\pm\lambda$height level loop of the
field. From the study of CLE4, Werner and me construct a time parameter
for each loop in CLE4 such that the time parameter transforms in a
conformally invariant way. From this construction of time parameter,
Sheffield, Watson and me derive the second coupling between GFF and
CLE4. In this talk, we will first give the background of SLE, CLE and
GFF. And then, discuss the coupling between GFF and SLE4, the first and
the second coupling between GFF and CLE4.
Friday, Nov 15, 2013

John Imbrie (University of Virginia)
Localization of eigenfunctions of random operators: A problem of percolation of resonances

Abstract: I will review what is known and expected for the Anderson
modelthe lattice Schroedinger equation with a random potential. For
strong disorder, eigenfunctions are localized, i.e. they decay
exponentially in the distance from a point. I will discuss a new proof
of this fact based on an iterative construction not unlike the Jacobi
diagonalization procedure. Convergence depends on the diluteness of
resonant regions in an associated correlated percolation problem.
Similar issues arise in the problem of localization for manybody
Hamiltonians with disorder. (Joint work with Tom Spencer).
Friday, Nov 22, 2013
Hairer, Junge, Nikeghbali, Williams
Friday, Dec 6 2013

Lingjiong Zhu (Morgan Stanley) [11am12noon]

Title: Nonlinear Hawkes Processes
Abstract: Hawkes process is a simple point process that has long
memory, clustering effect, selfexciting property and is in general
nonMarkovian. The future evolution of a selfexciting point process is
influenced by the timing of the past events. In this talk, we will
first discuss the limit theorems for linear and nonlinear Hawkes
processes. Then, we will drop the usual assumptions on Hawkes processes
and categorize it into different regimes, i.e. sublinear, subcritical,
critical, supercritical and explosive regimes. We will discuss time
asymptotics as well as some other properties.

Nicola Kistler (CUNY) [12noon1pm]

Title: Derrida's Generalized Random Energy Models  beyond spin glasses
Abstract: The random energy models, REMs for short, are simple Gaussian
fields with a builtin hierarchal structure. They have been introduced
by Bernard Derrida in the 80's to shed light on the mysterious Parisi
theory for mean field spin glasses. Within this context, the REMs have
provided invaluable insights ever since. On the other hand, recent
advances in disparate fields suggest that the REMs are truly universal
objects which play a fundamental role well beyond the original spin
glass setting. I will discuss the issue by means of a refinement of the
second moment method which is remarkably efficient in the study of the
extremes of large combinatorial structures whenever multiple scales can
be identified.