Schedule

• September 20

  Asaf Nachmias (MIT)

  Random walks on random fractals and the Alexander-Orbach conjecture

  Abstract: A simple random walk on the Euclidean lattice reaches distance ofabout n^{1/2} after n steps. On a (discrete) fractal, we expect therandom walker to spend most of its time on the "dangling ends" of thefractal and hence to slow down significantly. Alexander and Orbach(1982) conjectured that on fractals obtained from critical percolationon a lattice, the random walker reaches distance of about n^{1/3}after n steps. In this work we prove this conjecture when thedimension of the lattice is larger than 6.

  Based on joint work with Gady Kozma.

• September 27

  David Jerison (MIT)

  Internal Diffusion-Limited Aggregation.

  Abstract: Internal diffusion limited aggregation (IDLA) is a random growth model on the lattice defined for each integer time $n \geq 0$ by an {\bf occupied set} $A(n) \subset \mathbb Z^2$ as follows: begin with $A(0) = \emptyset$, $A(1) = \{0\}$, and then for each $n$add to $A(n)$ the first point at which a random walk from the originhits $\Z^2 \setminus A(n)$. IDLA was introduced by Meakinand Deutch in 1986 as a model for chemical processes suchas electopolishing, corrosion and etching. We discussjoint work with Lionel Levine and Scott Sheffield in which weshow that the deviation of $A(n)$from the disk is logarithmic in the radius, $r = \sqrt{n/\pi}$:There is an absolute constant $C$ such that almost surely forsufficiently large $n$,\[ B_{r - C\log r} \subset A(n) \subset B_{r+ C\log r}\]Moreover, the fluctuations can be described by a variant ofthe Gaussian Free Field. This allows us to confirmnumerical predictions made by Meakin and Deutch.

• October 4

  Horng-Tzer Yau (Harvard)

  Universality of random matrices and Dyson Brownian Motion.

  Abstract: Random matrices were introduced by E. Wigner to model the excitation spectrum of large nuclei.The central idea is based on the hypothesis thatthe local statistics of the excitation spectrum for a largecomplicated system is universalin the sense that it depends only on the symmetry classof the physical system but not on other detailed structures.Dyson Brownian motion is the flow of eigenvaluesof random matrices when each matrix element performs independentBrownian motions.In this lecture, we will explain the connection between theuniversality of random matrices andthe approach to local equilibrium of Dyson Brownian motion. The maintools in our approachare an estimate on the flow of entropy in Dyson Brownian motion anda local semicircle law.

• October 11

  Columbus Day.

• October 18

  Curtis McMullen (Harvard)

  Barycentric subdivision and random walks on the hyperbolic plane.

• October 25

  Jacob Fox (MIT)

  Dependent Random Choice.

  Abstract: We describe a simple and yet surprisingly powerful probabilistic technique which shows how to find in a dense graph a large subset ofvertices in which all (or almost all) small subsets have many commonneighbors. Recently this technique has had several striking applications toExtremal Graph Theory, Ramsey Theory, Additive Combinatorics, andCombinatorial Geometry. In this talk, which is based on a survey with BennySudakov, we discuss some of these applications.

• November 1

  Alexei Borodin (MIT)

  Growth of random surfaces

  Abstract: We describe a class of exactly solvable random growth models of oneand two-dimensional interfaces. The growth is local (distant parts of the interface growindependently), it has a smoothing mechanism (fractal boundaries do not appear),and the speed of growth depends on the local slope of the interface.The models enjoy a rich algebraic structure that is reflected through closeddeterminantal formulas for the correlation functions. Large time asymptoticanalysis of such formulas reveals asymptotic features of the emerging interfacein different scales. Macroscopically, a deterministic limit shape phenomenoncan be observed. Fluctuations around the limit shape range from universallaws of Random Matrix Theory to conformally invariant Gaussian processes inthe plane. On the microscopic (lattice) scale, certain universal determinantalrandom point processes arise.

• November 8

  Jason Miller (Microsoft research)

  CLE(4) and the Gaussian Free Field

  Abstract: The discrete Gaussian free field (DGFF) is the Gaussian measure on functions $h \colon D \to \R$, $D \subseteq \Z^2$ bounded, withcovariance given by the Green's function for simple random walk. Thegraph of $h$ is a random surface which serves as a physical model foran effective interface. We show that the collection of random loopsgiven by the level sets of the DGFF for any height $\mu \in \R$converges in the fine-mesh scaling limit to a family of loops which isinvariant under conformal transformations when $D$ is a latticeapproximation of a non-trivial simply connected domain. Inparticular, there exists $\lambda > 0$ such that the level sets whoseheight is an odd integer multiple of $\lambda$ converges to a nestedconformal loop ensemble with parameter $\kappa=4$ [so-called$\CLE(4)]$, a conformally invariant measure on loops which locallylook like $\SLE(4)$. Using this result, we give a coupling of thecontinuum Gaussian free field (GFF), the fine-mesh scaling limit ofthe DGFF, and $\CLE(4)$ such that the GFF can be realized as afunctional of $\CLE(4)$ and conversely $\CLE(4)$ can be made sense asa functional of the GFF. This is joint work with Scott Sheffield.

• November 15

  Fredrik Johansson Vilkund (Columbia University)

  Convergence rates for loop-erased random walk

  Abstract: Loop-erased random walk (LERW) is a self-avoiding random walk obtainedby chronologically erasing the loops of a simple random walk. In theplane, the lattice size scaling limit of LERW is known to be SLE(2), arandom fractal curve constructed by solving the Loewner differentialequation with a Brownian motion input.

  In the talk, we will discuss recent joint work with C. Benes (CUNY)and M. Kozdron (U. of Regina) on obtaining a rate for the convergenceof LERW to SLE(2). More precisely, we will outline our derivation of arate for the convergence of the Loewner driving function for LERW toBrownian motion with speed 2 on the unit circle, the Loewner drivingfunction for SLE(2).

  We will then show how to use this to obtain a rate for the convergenceof the paths with respect to Hausdorff distance. Time permitting, wewill also indicate how some of these results can be extended tocertain other models known to converge to SLE.

• November 22

  Tim Austin (Brown University)

  Compression exponents for finitely generated groups.

  Abstract: One key insight behind geometric group theory is that some algebraic properties of infinite discrete groups can be understood byconsidering them more coarsely as discrete metric spaces withtheir word metrics. One invariant for such groups arising in thisway is their compression exponent, an indicator of how badly thatmetric must be distorted if the group is embedded into various classesof Banach space. I will review this definition and some results relating itto the algebraic structure of the group, and then discuss someexamples of groups for which the exact values of these exponents areknown, and in particular the (usually less obvious) arguments that gointo bounding them from above (i.e., 'putting an upper bound on thequality of an arbitrary embedding').

• November 29

  Clement Hongler (Columbia University)

  Critical Ising crossing probabilities and SLE

  Abstract: We study the Ising model at criticality from an SLE point of view. The interfaces between + and - spins of the Ising model with Dobrushin+/- boundary conditions have been shown to converge to SLE(3) bySmirnov (on the square lattice) and Chelkak and Smirnov (on moregeneral lattices), thanks to the introduction and proof of convergenceof a discrete holomorphic martingale observable in this setup.We show conformal invariance of the Ising interfaces in presence offree boundary conditions. In particular we prove the conjecture ofBauer, Bernard and Houdayer about the scaling limit of interfacesarising in a so-called dipolar setup. The limiting process is aLoewner chain guided by a drifted Brownian motion, known as dipolarSLE or SLE(3,-3/2) in the literature.This case is made harder by the absence of natural discreteholomorphic martingales, requiring us to introduce "exotic"martingale observables. The study of these observables is performed byKramers-Wannier duality and Edwards-Sokal coupling, and thecomputation of the scaling limit is made by appealing to discretecomplex analysis methods, to three existing convergence results aboutdiscrete fermions, to the scaling limit of critical Fortuin-Kasteleynmodel interfaces and to the introduction of Coulomb gas integrals.Our result allows to show conjectures by Langlands, Lewis andSaint-Aubin about conformal invariance of crossing probabilities forthe Ising model.Based on joint work with Kalle Kytölä and work in progress with HugoDuminil-Copin.

• December 6

  James Lee (University of Washington)

  Cover times, blanket times, and the Gaussian free field

  Abstract: The cover time of a finite graph (the expected time for the simplerandom walk to visit all the vertices) has been extensively studied,yet a number of fundamental questions concerning cover times haveremained open. Winkler and Zuckerman (1996) defined the blanket time(when the empirical distribution if within a factor of 2, say, of thestationary distribution) and conjectured that the blanket time isalways within O(1) of the cover time. Aldous and Fill (1994) askedwhether there is a deterministic polynomial-time algorithm thatcomputes the cover time up to an O(1) factor. The best approximationfactor found so far for both these problems was (log log n)^2 forn-vertex graphs, due to Kahn, Kim, Lovasz, and Vu (2000).

  We show that the cover time of a graph, appropriately normalized, isproportional to the expected maximum of the (discrete) Gaussian freefield on G. We use this connection and Talagrand's majorizingmeasures theory to deduce a positive answer to the question of Aldousand Fill and to establish the conjecture of Winkler and Zuckerman.These results extend to arbitrary reversible finite Markov chains.

  This is joint work with Jian Ding (U. C. Berkeley) and Yuval Peres(Microsoft Research).