Random Explorations

We consider a uniform spanning tree under an n-arm event. The total winding of the branches has an exact asymptotic formula, which involves the Brownian loop measure and -- interestingly -- depends on the total number of branches only through its parity. Not surprisingly, the branches converge to multiradial SLE(2), and the law of the hitting points of the branches to the boundary coincides with that of eigenvalues of an COE random matrix.

Joint work with Nathanael Berestycki, Marcin Lis, and Mingchang Liu.

A classical topic of study concerns points that have been visited exceptionally often by a 2d simple random walk or Brownian motion. I will explain how this study can lead to the definition of new conformally invariant fields, generalising random walk representations of the Gaussian free field. These fields are built out of a Brownian loop soup and are closely related to the structure of clusters.

The conformal dimension of a metric space $(X,d)$, introduced by Pansu in 1989 in the context of Gromov-hyperbolic groups, is the infimum of the Hausdorff dimensions of all metric spaces quasisymmetrically equivalent to $(X,d)$. It is always bounded between the topological and Hausdorff dimensions of $(X,d)$. In this talk, we describe recent work showing that the conformal dimension of the Brownian sphere is exactly 2, matching its topological dimension. We also discuss a complementary result establishing that the Brownian sphere is quasisymmetrically rigid, meaning its group of quasisymmetric automorphisms is almost surely trivial. This talk is based on joint work with Yi Tian.

This is joint work with Catherine Wolfram. We describe a new family of determinantal probability measures in d dimensions called Independent Coverings. Given a region G in Z^d of size n, and fixed polyomino t, an independent covering of G by t is collection of n different translates of t which covers G and is independent in the sense that the incidence matrix between those translates and G is of full rank.  We define a natural probability measure on the set of independent coverings, show that it is a determinantal measure, and compute some of its properties, including scaling limits and, sometimes, conformal invariance.   Generalizations to multiple tiles are readily available.

We present an upper bound for the rate of convergence of the random walk Green's function to the Brownian motion Green's function in cone-convex domains. The bound unsurprisingly depends on the regularity of the domain.

We introduce a family of random closed subsets of Brownian traces, called Brownian loop catchers, with central charge in the range $-2le c<0$. Such a Brownian loop catcher with central charge $c$, together with the loops intersecting it from an independent Brownian loop soup of intensity $-c/2$, recovers the original Brownian trace in law. At $c=-2$, this object is the continuum loop-erased random walk, which is SLE$_2$ in two dimensions; as $cuparrow 0$, it degenerates to the Brownian trace itself. We prove that in two dimensions, the outer boundary of a Brownian loop catcher with central charge $-2le c<0$ is locally SLE$_kappa$, where $c=1-6left(frac{2}{sqrt{kappa}}-frac{sqrt{kappa}}{2}right)^2$. Consequently, a planar Brownian trace contains an SLE$_kappa$-type curve for every $kappain[2,8/3]$.

We will discuss conditions on Sierpinski carpets that allow for uniformization to round carpets, with an eye towards application to the conformal loop ensemble.

I will discuss spin models on Z^2 taking values in either a sphere or a real projective space. Understanding the scaling limits of these models (or even understanding whether one has exponential decay of correlations at low temperatures) is a major open problem. I will explain how understanding this model is in some sense equivalent to understanding a natural "random loop ensemble" model, related to loop soups. I will discuss both what is known and what remains to be understood.

It is well known that the non-intersection exponents for planar Brownian motion agree with the alternating (2N)-arm exponents for 2D percolation, both of which were originally predicted in the physics literature using conformal field theory (CFT). For 2D percolation, the CFT perspective has led to a range of exact solvability results going far beyond scaling exponents. This motivates a systematic study of planar Brownian motion from the same viewpoint.

In this talk, I will present recent progress in this direction, including several new exact formulae for Brownian partition functions in the annulus. The talk is based on joint works with Xuesong Fu, Gefei Cai and Zhuoyan Xie; with Gefei Cai, Jiaqi Liu, Wei Qian and Baojun Wu; with Guillaume Baverez, Antoine Jego and Baojun Wu; and with Gefei Cai, Baojun Wu and Zhuoyan Xie.

Starting from a simple observation relating the Brownian loop soup and SLE loop measure, and exploiting the connections between the Brownian loop soup, Liouville quantum gravity, and Liouville conformal field theory, we derive the joint law of the outer and inner boundaries of a Brownian loop-soup cluster with respect to a given point. In particular, we compute the probability that the two boundaries do not touch. We also extend this result to a single Brownian loop by taking the zero-intensity limit of the loop soup. This is based on a joint work with Gefei Cai, Jiaqi Liu, Xin Sun and Baojun Wu.