Beijing Institute of Mathematical Sciences and Applications

Given n uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them ? "Fairly" means that each region has the same area. It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting gradient flow yield such a partition—with exactly equal areas, no matter how the points are distributed. (See the cover of the AMS Notices at http://www.ams.org/publications/journals/notices/201705/rnoti-cvr1.pdf or the PNAS article http://www.pnas.org/content/early/2018/09/06/1720804115 ). Our main result is that this partition minimizes, up to a bounded factor, the average distance between points in the same cell. I will also present an application to almost optimal matching of n uniform blue points to n uniform red points on the sphere, connecting to a classical result of Ajtai, Komlos and Tusnady (Combinatorica 1984). I will emphasize open problems on the diameters of the basins and the behavior of greedy matching schemes. Joint work with Nina Holden and Alex Zhai.

The development of stochastic differential equations has a long history. In recent years, the study of singular SDEs is attracted much attention not only in mathematics, but also in applications. In this talk I will survey some recent results about singular stochastic differential equations including the McKean-Vlasov SDEs.

In this talk, I will introduce some fundamental ideas in ergodic Ramsey theory. The Furstenberg-Sárközy theorem asserts that the difference set E-E of a subset E of the natural numbers with positive upper density contains a (nonzero) square. Furstenberg's approach relies on a correspondence principle and a version of the Poincaré recurrence theorem along squares; the latter is shown via the result that for any measure-preserving system $(\Omega,\mathcal{A},\ mathbf{P},T)$ and set A with positive measure, the ergodic average $\frac{1}{N} \sum_{n=1}^N \ mathb{P}(A \cap T^{-n^2}A)$ has a positive limit c(A) as N tends to infinity. Motivated -- by what? we shall see -- to optimize the value of c(A), we define the notion of asymptotic total ergodicity in the setting of modular rings $\mathbb{Z}/N\mathbb{Z}$. We show that a sequence of modular rings (Z/N_m Z) is asymptotically totally ergodic if and only if the least prime factor of N_m grows to infinity. From this fact, we derive some combinatorial consequences. These results, and some extensions we will, time-permitting, also discuss, are based on joint work with Vitaly Bergelson.

For the Ising model defined on a finite domain $D$ at temperature $T$ and external field $h$, let $\alpha_1(D, T)$ be the modulus of the first zero (that closest to the origin) of its partition function. We prove that $\alpha_1(D, T)$ decreases to $\alpha_1(T)$ as $D$ increases to the whole space where $\alpha_1(T)$ is the radius of the largest disk centered at the origin in which the free energy in the thermodynamic limit is analytic. We also note that $\alpha_1(T)$ is strictly positive if and only if $T$ is strictly larger than the critical temperature. Based on joint work with Federico Camia and Charles M. Newman.

We formulate a new criterion of the asymptotic stability for some non-equicontinuous Markov semigroups, the so-called eventually continuous semigroups. We further provide a class of place dependent iterated function systems with essential randomness, which are non-equicontinuous Markov semigroups. We also present an approach to prove the unique ergodicity of a large class of SPDEs driven by multiplicative noise.

In this talk, I will discuss Lévy type operators and some regularity estimates for solutions to corresponding equations. Our emphasis is on kernels with a critically low singularity which does not allow for standard scaling. For example, I will talk about operators that have a logarithmic order of differentiability.

The central limit theorem is one of the most important results in probability, and it has many generalizations in random processes and other stochastic models. In the last decades, the heat kernel estimates on percolation cluster and other random conductance models have been largely studied and many results of CLT type are obtained. In this talk, I will review these results and present a new heat kernel estimate, which can be seen as a quantitative local central limit theorem on the infinite percolation cluster. This talk is based on a joint work with Paul Dario.

Abstract: The theory of Gaussian Multiplicative Chaos played a key role in the recent development on the probabilistic study of the Liouville conformal field theory. However, the origin of the theory partially comes from Kolmogorov’s theories of turbulence. I will introduce the origin of the theory of Gaussian Multiplicative Chaos, its modern revisions and some of its broad applications to other branches of probability and mathematics, based on my personal knowledge about this vast subject.

We consider a discrete-time branching simple random walk on Z^d, where each particle independently makes simple random walk and produces a random number of children so that the offspring law is of mean 1 and of finite variance. If the system starts from some far away position x, we are wondering how many particles could hit the origin.

A composition of a positive integer $n$ is a sequence of positive integers that sum to $n$. In this talk, I will introduce a family of interval-partition-valued diffusions that arise as limits of random walks on integer compositions. These infinite-dimensional diffusions have the self-similarity and branching property. Our model is closely related to Pitman--Dubins Chinese restaurant processes and random walks on integer partitions studied by Borodin--Olshanski and Petrov. I will also talk about some applications of our model in population genetics and continuum-tree-valued dynamics. This talk is based on joint work with Noah Forman, Douglas Rizzolo, and Matthias Winkel.

We consider the additive martingale $W_t(\lambda)$ and the derivative martingale $\partial W_t(\lambda)$ for one-dimensional supercritical super-Brownian motions with general branching mechanism. In the critical case $\lambda=\lambda_0$, we prove that $\sqrt{t}W_t(\lambda_0)$ converges in probability to a positive limit, which is a constant multiple of the almost sure limit $\partial W_\infty(\lambda_0)$ of the derivative martingale $\partial W_t(\lambda_0)$. We also prove that, on the survival event, $\limsup_{t\to\infty}\sqrt{t}W_t(\lambda_0)=\infty$ almost surely.

In the shotgun assembly problem for a graph, we are given the empirical profile for rooted neighborhoods of depth $r$ (up to isomorphism) for some $r \geq 1$ and we wish to recover the underlying graph up to isomorphism. When the underlying graph is an Erdős-Rényi $\mathcal{G}\left(n, \frac{\lambda}{n}\right)$, we show that the shotgun assembly threshold $r_{*} \approx \frac{\log n}{\log \left(\lambda^{2} \gamma_{\lambda}\right)^{-1}}$ where $\gamma_{\lambda}$ is the probability for two independent Poisson-Galton-Watson trees with parameter $\lambda$ to be rooted isomorphic with each other. Our result sharpens a constant factor in a previous work by Mossel and Ross (2019) and thus solve a question therein. This talk is based on a joint work with Jian Ding and Yiyang Jiang.

Consider a general class of random band matrices $H$ on the $d$-dimensional lattice of linear size $L$. The entries of $H$ are independent centered complex Gaussian random variables with variances $s_{xy}$, which have a banded profile so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. In dimensions $d\ge 7$, assuming that $W\geq L^\delta $ for a small constant $\delta>0$, we prove the deloclaization and quantum unique ergodicity (QUE) of the bulk eigenvectors of $H$. Furthermore, we prove the bulk universality of $H$ under the condition $W \gg L^{95/(d+95)}$. In the talk, I will discuss a new idea for the proof of the bulk universality through QUE, which verifies the conjectured connection between QUE and bulk universality. The proof of QUE is based on a local law for the Green's function of $H$ and a high-order $T$-expansion developed recently. Based on Joint work with Xu, Yau and Yin.

Conformal invariance of critical lattice models in two-dimensional has been vigorously studied for decades. In this talk, we focus on connection probabilities of critical random cluster model in polygons with alternating boundary conditions. This talk has two parts. In the first part, we consider critical random-cluster model with cluster weight $q\in (0,4)$ and introduce conjectural formulas for connection probabilities of multiple interfaces. In the second part, we give rigorous results for $q=2$, i.e. the FK-Ising model. This talk is based on a joint work with Eveliina Peltola and Hao Wu.