# SCIENTIFIC PROGRAMS AND ACTIVITIES

May 24, 2015
THE FIELDS INSTITUTE
FOR RESEARCH IN MATHEMATICAL SCIENCES
 Toronto Probability Seminar 2011-12 held at the Fields Institute, Stewart Library Mondays at 4:10 p.m. Organizers Bálint Virág , Benedek Valkó University of Toronto, Mathematics and Statistics For questions, scheduling, or to be added to the mailing list, contact the organizers at: probsem-at-math-dot-toronto-dot-edu
 See Seminar Home page for updates Archive of 2010-11 Talks Map to Fields Archive of 2009-10 Talks
 PAST SEMINARS 2011-2012 Friday, May 4 3:10 pm Lukasz Grabowski (Imperial College, London) Decidability aspects of computing spectral measures Given a finitely generated group G we can fix a generating set g_1, g_2, ... g_n and consider T to be a random walk (or more general convolution operator) on the Cayley graph of G wrt the generators g_1, ..., g_n. In the talk we will investigate computational problems related to computing the spectral measure of T: in particular, is there an algorithm which answers the question is the kernel of T non-trivial?" I will give many examples of groups where there is such an algorithm and sketch a proof why there is no such algorithm for the group $H^4$, where H is the lamplighter group $Z_2 \wr Z$. May 7 Alejandro Ramirez (Pontificia Universidad catolica de Chile) Criteria for ballistic behavior of random walks in random environment It is conjectured that for a random walk on the hipercubic lattice Z^d in a uniformly elliptic i.i.d. random environment, when the dimension d ? 2, transience in a given direction implies ballisticity in the same direction. In this talk, I will review some recent progress on this question made in collaboration with Noam Berger, David Campos and Alexander Drewitz. In particular, I will introduce a set of polynomial ballisticity criteria and several renormalization methods. April 16 Fields Institute Room 230 Janosch Ortmann (University of Warwick) Product-form invariant measures for Brownian motion with drift satisfying a skew-symmetry type condition Motivated by recent developments on positive-temperature polymer models we propose a generalisation of reflected Brownian motion (RBM) in a polyhedral domain. Our process is obtained by replacing the singular drift on the boundary by a continuous one which depends, via a potential U, on the position of the process relative to the domain. We show that our generalised process has an invariant measure in product form if we have a certain skew-symmetry condition that is independent of the choice of potential. Applications include a exponential analogue of the Brownian TASEP, examples motivated by queueing theory, Brownian motion with rank-dependent drift and a process with close connections to the \delta-Bose gas. April 2 Benedek Valko (UW-Madison) Scaling exponents of lattice gases We consider a a family of lattice gas models (speed change models) where the particles move randomly according to a local rule with the extra condition that there is at most one particle at any site. We study the equilibrium fluctuations at a given particle density, in particular we are interested in the scaling exponents of certain physical quantities. These exponents are predicted to be governed by the local behavior of the macroscopic flux function at the equilibrium density. We will give upper and lower bounds on these exponents, in particular we will confirm the superdiffusive behavior of the models in all the cases where this was predicted by physical arguments. Joint with Jeremy Quastel (Toronto). March 14 3:10 p.m Perla Sousi (University of Cambridge) The effect of variable drift on Brownian motion and the Wiener sausage The Wiener sausage at time t is the algebraic sum of a Brownian path on [0,t] and a ball.Does the expected volume of the Wiener sausage increase when we add drift? How do you compare the expected volume of the usual Wiener sausage to one defined as the algebraic sum of the Brownian path and a square (in 2D) or a cube (in higher dimensions)? We will answer these questions using their relation to the detection problem for Poisson Brownian motions, and rearrangement inequalities on the sphere. (Talk based on joint works with Yuval Peres) March 16 CANCELLED: Van Vu (Yale University) March 5 James Nolen (Duke University) Normal approximation for a random elliptic PDE I will talk about solutions to an elliptic PDE with conductivity coefficient that varies randomly with respect to the spatial variable. It has been known for some time that homogenization may occur when the coefficients are scaled suitably. Less is known about fluctuations of the solution around its mean behavior. For example, if an electric potential is imposed at the boundary, some current will flow through the material. What is the net current? For a finite random sample of the material, this quantity is random. In the limit of large sample size it converges to a deterministic constant. I will describe a central limit theorem: the probability law of the energy dissipation rate is very close to that of a normal random variable having the same mean and variance. I'll give an error estimate for this approximation in total variation. Feb. 27 Yuri Bakhtin (Georgia Tech) Randomly forced Burgers equation in noncompact setting The Burgers equation is one of the basic hydrodynamic models that describes the evolution of velocity fields of sticky dust particles. When supplied with random forcing it turns into an infinite-dimensional random dynamical system that has been studied since late 1990's. The variational approach to Burgers equation allows to study the system by analyzing optimal paths in the random landscape generated by random force potential. Therefore, this is essentially a random media problem. For a long time only compact cases of Burgers dynamics on the circle or a torus were understood well. In this talk, I will discuss the quasi-compact case where the random forcing decays to zero at infinity and the completely noncompact case of forcing that is stationary in space-time. The main result is the description of the ergodic components for the dynamics and One Force One Solution principle on each of the components. Joint work with Eric Cator and Kostya Khanin. Feb. 15 5:00 p.m *Please not non-standard date and time Charles Bordenave (Université de Toulouse) Localization and delocalization of eigenvectors for heavy-tailed random matrices This is a joint work with Alice Guionnet. Consider an n x n Hermitian random matrix with, above the diagonal, independent entries with alpha-stable symmetric distribution and 0 < alpha < 2. The spectrum of this random matrix differs significantly from the spectrum of Wigner matrices with finite variance. It can be seen as an instance of a sparse random matrix : only O(1) entries in each row have a significant impact on the behavior of the matrix. In this talk, we will give bounds on the rate of convergence of the empirical spectral distribution of this random matrix as n goes to infi nity. When 1 < alpha < 2 and p large enough, we will see that the Lp-norm of the eigenvectors normalized to have unit L2-norm goes to 0. On the contrary, when 0 < alpha < 2/3, we will see that these eigenvectors are localized. These localization/delocalization results only partially recover some predictions due to Bouchaud and Cizeau in 1994. Feb. 6 Daniel Remenik (University of Toronto) Variational formulas for the Airy2 process The Airy2 process arises as the scaling limit of the fluctuations in a variety of one dimensional random growth models with curved initial conditions and point-to-point directed random polymers, all in the KPZ universality class. In this talk I will explain how to get a formula for the continuum statistics of the Airy2 process or, more precisely, for the probability that the process lies below a given function on some finite interval. Then I will explain how this formula can be used to confirm some predictions which relate a variational problem for the Airy2 process to certain asymptotic distributions in the KPZ universality class associated with other initial conditions. I will also explain how the formula can be used to compute the asymptotic endpoint distribution for point-to-line polymers. This is joint work with Ivan Corwin, Gregorio Moreno Flores and Jeremy Quastel. Jan. 30 Karl Liechty (University of Michigan) Nonintersecting processes on the half-line and discrete orthogonal polynomials We will consider a system of n Brownian motions on the time interval [0,1] taking values the nonnegative real numbers such that all of them begin at zero at time t=0, and return to zero at time t=1, and such that they never intersect for t \in (0,1). The condition that the Brownian motions are never negative can be thought of as a "wall" at zero. In the proper scaling, the distribution of the top curve should converge to the Airy process, and thus the distribution of the maximum value of the top curve should converge to the Tracy-Widom distribution for the Gaussian orthogonal ensemble. I will formulate the distribution of the maximum value of the top curve in terms of a system of discrete orthogonal polynomials which then can be evaluated in an appropriate double scaling limit. This double scaling limit corresponds to the system of orthogonal polynomials approaching saturation, meaning that their zeroes become as tightly packed as possible. If time permits, I will also discuss a discrete version of the same problem. Jan. 23 3:10 p.m   **Please note non-standard time Raoul Normand (Paris VI, Toronto) Self-organized criticality in a discrete model of coagulation The goal of this talk is to present a random model of coagulation, meant to be a discrete model for Smoluchowski's equation. Loosely, one starts with a large number of particles, each with a certain number of arms, used for the coagulations. Then we pair, at each step, two arms chosen uniformly at random, but only in "small" clusters. We want to study the shape of the clusters in this model, and we will explain how it exhibits a phenomenon of self-organized criticality. This is a joint work with Mathieu Merle (LPMA, Univ. Paris VII). Jan.16 3:10 p.m Almut Burchard (University of Toronto) Random sequences of symmetrizations In geometric optimization problems, it can be useful to approximate the symmetric decreasing rearrangement by a sequence of simpler rearrangements, such as polarization (two-point rearrangement), Steiner symmetrization, or the Schwarz rounding process. In the literature, considerable effort goes into the construction of special sequences of rearrangements that yield full rotational symmetry in the limit. Here, I will discuss conditions under which random sequences of rearrangements converge almost surely to the symmetric decreasing rearrangement and give bounds on the rate of convergence. I will also show examples how convergence can fail. The talk is based on results from Marc Fortier's 2010 M.S. thesis and more recent results, including work in progress with Gabriele Bianchi, Paolo Gronchi, and Ajosa Volcic. Jan. 9 3:10 p.m Codina Cotar (Fields Institute) Uniqueness of random gradient states We consider two versions of random gradient models. In model A) the interface feels a bulk term of random fields while in model B) the disorder enters though the potential acting on the gradients itself. It is well known that without disorder there are no Gibbs measures in infinite volume in dimension d = 2, while there are gradient Gibbs measures describing an infinite-volume distribution for the increments of the field, as was shown by Funaki and Spohn. Van Enter and Kuelske proved that adding a disorder term as in model A) prohibits the existence of such gradient Gibbs measures for general interaction potentials in d = 2. Cotar and Kuelske proved the existence of shift-covariant gradient Gibbs measures for model A) when d\ge 3 and the expectation with respect to the disorder is zero, and for model B) when d\ge 2. In the current work, we prove uniqueness of shift-covariance gradient Gibbs measures with given tilt under the above assumptions. (this is joint work with Christof Kuelske). Dec. 16 3:10 p.m Christian Sadel (UC Irvine) A multi-channel 1D random Dirac operator with purely absolutely continuous spectrum For Anderson-type random operators one expects the existence of absolutely continuous spectrum for low disorder in 3 and higher dimensions. In contrast, one-dimensional structures usually have pure point spectrum because the Lyapunov exponents are positive. However, certain symmetries in the model can lead to zero Lyapunov exponents and absolutely continuous spectrum. We will give such an example in the form of a multi-channel Dirac operator with random matrix potential with time-reversal symmetry. Dec. 9 3:10 p.m Robert Young (University of Toronto) Pants decompositions of random surfaces Random graphs often have remarkable geometric and combinatorial properties, but what about random high-genus surfaces? In this talk, I will describe a construction of a random high-genus surface and prove one of its unusual geometric properties. No previous knowledge of the geometry of surfaces is necessary. This is joint work with Larry Guth and Hugo Parlier. Dec. 2 3:10 p.m Arnab Sen, Cambridge University Reverse hypercontractivity and its applications An operator T is said to satisfy a hypercontractive inequality if there exist q > p > 1 (depending on T) such that |Tf|_q \leq \|f\|_p for all functions f. Hypercontractive inequalities are extremely well known and play a fundamental role in discrete harmonic analysis as well as other areas of mathematics. An operator T is said to satisfy a reverse hypercontractive inequality if there exist q < p < 1 (q and p can be negative) such that |Tf|_q \geq |f|_p for all strictly positive functions f. Reverse hypercontractive inequalities started to emerge in recent years as a useful tool for providing solutions to a number of problems. In this talk I will discuss some new results relating reverse hypercontractive inequalities to hypercontractive, Log-Sobolev and Poincare inequalities in the setting of finite reversible Markov chains. I will also describe some of the old and new applications of the reverse hypercontractive inequalities. This is joint work with Elchanan Mossel and Krzysztof Oleszkiewicz. ******************************************** 2:00 p.m Probability Study Group BA 6180 Raoul Normand, Paris 6, Toronto A model of migration under constraints The goal of this talk is to present a simple model of population with migration. The tools used are classical, namely Galton-Watson trees, random walks and Brownian motion, so this talk is accessible to graduate students. In our model, migrations are constrained, in that people migrate when the resources on the "island" where they live are exhausted. From the genealogy of an individual (a Galton-Watson tree), we can construct the "tree of isles", describing the genealogy of the migrations. A first step is to describe this tree. The second step is to study limits. The relevant parameters are encoded in a random walk, and thus the limiting quantities are related to the Brownian motion. Nov. 25 3:10 p.m. Carl Mueller, Rochester Uniqueness and nonuniqueness for some stochastic PDE Good uniqueness criteria for stochastic differential equations have been known for a long time, at least in the one dimensional case. The situation is very different for stochastic PDE (SPDE), and uniqueness criteria for parabolic SPDE have only appeared recently. We will discuss some of this progress, focusing on equations related to the superprocess, which is a limit of branching Brownian motions. Nov. 18 2:10 pm Room 6180, Bahen *Please note non-standard time and location Affliated Actvity Probability Study Gruop Codina Cotar will give the first of three lectures on Gradient Models Nov. 18 3:10 pm Joseph Najnudel (Zurich) A unitary extension of virtual permutations The virtual permutations, introduced by Kerov, Olshanski and Vershik, are sequences of permutations of increasing order, whose cycle structure satisfies some compatibility properties. If these permutations are chosen uniformly and considered as random matrices, then one can prove an almost sure convergence of their renormalized eigenangles. In a joint paper with P. Bourgade and A. Nikeghbali, we extend this result to sequences of matrices following Haar measure on the unitary group. Nov. 16 3:10 pm *Please note non-standard time Jeremy Quastel (University of Toronto) Variational problems for Airy processes We show how to compute the joint density of the max and argmax of the Airy_2 process minus a parabola. The argmax is a universal distribution governing the endpoint of directed random polymers in 1+1 dimensions. Nov. 11 2:30 pm *Please note non-standard time Thomas Bloom (University of Toronto) Large deviations for random matrices via potential theory Ben Arous and A.Guionnet gave the first Large Deviation result for the Gaussian Unitary Ensemble.(drawing on work of Voiculescu). Their method was subsequently extended to general Unitary and other ensembles.I will outline a new proof, using potential theory, of those large deviation results. I will also discuss a large deviation result for a multivariable generalization of Unitary ensembles. (This uses recent developments in pluripotential theory due to R.Berman and S.Boucksom.) No prior knowledge of potential or pluripotential theory will be assumed. Nov. 4 3:10 pm Robert McCann (Toronto) Imagine some commodity being produced at various locations and consumed at others. Given the cost per unit mass transported, the optimal transportation problem is to pair consumers with producers so as to minimize total transportation costs. Despite much study, surprisingly little is understood about this problem when the producers and consumers are continuously distributed over smooth manifolds, and optimality is measured against a cost function encoding some geometry of the product space. This talk will include an introduction to the optimal transportation, its relation to Birkhoff's problem of characterizing of extremality among doubly stochastic measures, and some recent progress linking the two. In particular, we expose the topology of the cross-difference, which explains why extremal measures concentrate on thin sets, and which underlies a criterion for uniqueness of solutions subsuming all previous criteria, and which is among the very first to apply to smooth costs on compact manifolds, yet remains limited to topological spheres. ***Probability Study Group 2:00-3:00 pm BA6180. Oct. 28 3:10 pm Ben Rifkind (Toronto) A Stochastic Version of Multiplicative Cascades I will discuss a 1 dimensional version of a random geometry know as Multiplicative Cascades. In 2008, Benjamini and Schramm proved a KPZ relation which describes this random geometry relative to the regular Euclidean one. In joint work with Tom Alberts, we study the evolution of this random geometry in time and extract an ODE for the KPZ formula. Oct. 21 3:10 pm Balint Virag (Toronto) Finite graphs and Kesten's theorem Kesten showed that any transitive infinite d-regular graph whose spectral radius is minimal is a tree. In this talk, we present a version of this theorem for general finite d-regular graphs: if the spectral radius is close to minimal, then a large neighborhood around most points is a tree. This is a joint work with M. Abert and Y. Glasner. BA6290B (small meeting room across from the math library) from 2-3 on Friday. Let I_N denote the size of the largest independent set of the Erdos-Renyi random graph G(N,cN) consisting of N vertices and chosen uniformly at random from the set of all graphs on N vertices with cN edges. It was conjectured that I_N/N converges to a limit almost surely, and this was recently proved by Bayati, Gamarnik and Tetali using a clever combinatorial interpolation argument. I will present their argument, which was also used to prove almost sure scaling limits for other combinatorial problems on random graphs. ------------------------------ ***Probability Study Group 2:00 - 3:00 p.m. *Please note this event is at the Bahen Centre BA6290B (small meeting room across from the math library) Let I_N denote the size of the largest independent set of the Erdos-Renyi random graph G(N,cN) consisting of N vertices and chosen uniformly at random from the set of all graphs on N vertices with cN edges. It was conjectured that I_N/N converges to a limit almost surely, and this was recently proved by Bayati, Gamarnik and Tetali using a clever combinatorial interpolation argument. I will present their argument, which was also used to prove almost sure scaling limits for other combinatorial problems on random graphs. Oct. 14 3:10 pm Narn-Rueih Shieh (Taipei) Scaling Limits for some PDEs with Random Initial Data Let X(x,t) (for x in R^n and t in R+) be the spatial-temporal random field arising from a certain parabolic PDE with initial data given by a subordinated Gaussian field. We discuss the scaling limit of such a space-time random field. The space-time correlation structure of the solution field and the Hermite expansion associated with the initial data play the essential roles in our study. Similar results hold for X an R^2-valued spatial-temporal random field arising from a certain two-equation system, under very weak coupling. Scaling limits for time-fractional and spatial-fractional systems are also reported. There are some novel features in the fractional case. This talk is based on joint work with G.-R. Liu (a PhD student in Taiwan) Sept. 12 3:10 pm Balazs Szegedy Graph limits and corresponding spectral theory The foundation of the so-called graph limit theory is a certain compactifiction of the set of finite graphs which captures both local (weak) and global (Szemeredi partition) convergence. The purpuse of this talk is to extend the spectral theory of finite graphs to the graph limit space. Along these lines we prove a spectral version of Szemeredi's regularity lemma.