Panelists:

Shige Peng

• Professor of Mathematics, Institute of Mathematics, Shandong University
• Academician of the Chinese Academy of Sciences

Qi-Man Shao

• Chairman and Choh-Ming Li Professor of Statistics, Department of Statistics, The Chinese University of Hong Kong

Zhan Shi

• Professor, Université Paris VI

Vladas Sidoravicius

• Professor of Mathematics at NYU and NYU Shanghai

Zhonggen Su

• Professor, Department of Mathematics, Zhejiang University

Jiangang Ying

• Professor, School of Mathematical Sciences, Fudan University

Wei’an Zheng

• Professor/ Chair, Department of Financial Engineering, East China Normal University

*Alphabetically Ordered by Surname

By Fengyu WANG, Beijing Normal University

By using the  local  dimension-free Harnack inequality established on incomplete Riemannian manifolds, integrability conditions on the coefficients are presented for SDEs to imply the non-explosion of solutions  as well as the  existence, uniqueness and regularity estimates of invariant probability measures. These conditions include a class of drifts unbounded on compact domains such that the usual Lyapunov conditions can not be verified. The main results are extended to second order differential operators on Hilbert spaces and  semi-linear SPDEs.

By Roberto Fernandez, Utrecht University

Non-Gibbsian measures were initially studied in relation to renormalization transformations.  In this setting they emerged via phase transitions in the ensemble of original (internal) spins conditioned to a particular renormalized configuration. Later, the same mathematical techniques led to proofs that low-temperature Ising measures subjected to a high-temperature spin-flip evolution can become non-Gibbsian after a finite time. These proofs, however, amount to a "static" view of the evolution as projections of systems with one additional (time) dimension. This was an unsatisfactory state of affairs that did not seem to lead to a truly dynamical understanding of the onset of non-Gibbsianness. As a response to this criticism, in the last few years a new paradigm has been developed in which Gibbs-non-Gibbs transitions are related to changes in the large-deviation rates of conditioned evolutions of measures: Rates with multiple global minima lead to non-Gibbsianness. I will present the main ideas behind this new approach and report on rigorous results for mean-field and local mean-field models.

By Federico Camia, NYU Abu Dhabi

Several classical lattice models of statistical mechanics, such as percolation and the Ising and Potts models, can be described in terms of clusters. In the last fifteen years, there has been tremendous progress in the study of the geometric properties of such models in two dimensions in the scaling limit, when the lattice spacing is sent to zero. Much of that work has focused on cluster boundaries, using the Schramm-Loewner Evolution (SLE), introduced by Oded Schramm, and collections of SLE loops called Conformal Loop Ensembles (CLEs). In this talk I will discuss the scaling limit of the clusters themselves and their ''areas'' in the case of percolation and the Ising model. This leads to the study of rescaled counting measures and to the concept of Conformal Measure Ensembles (first introduced in joint work with Chuck Newman), with interesting applications to two-dimensional critical percolation and the two-dimensional critical Ising model.

Based on joint work with Rene Conijn and Demeter Kiss.

By Nike SUN, University of California, Berkeley

We establish the random k-SAT threshold conjecture for all k exceeding an absolute constant k(0). That is, there is a single critical value c*(k) such that a random k-SAT formula at clause-to-variable ratio c is with high probability satisfiable for c < c*(k), and unsatisfiable for c > c*(k). The threshold c*(k) matches the explicit prediction derived by statistical physicists on the basis of the so-called "one-step replica symmetry breaking" (1RSB) heuristic. In the talk I will describe the main obstacles in computing the threshold, and indicate how they are overcome in our proof.

Joint work with Jian Ding and Allan Sly.

By Wei WU, NYU Shanghai

The minimal spanning tree model has been widely applied to combinatorial optimizations and to the study of disordered physical systems. For the infinite lattice $\mathbb{Z}^d$, rigorous results for the geometry of the minimal spanning forest were recently proved for $d= 2$ and still  remain open for $d \geq 3$. We made partial progress by proving that the minimal spanning forest measure is supported on a single tree for quasi-planar graphs, such as the two dimensional slabs. Our proof uses the connections between the minimal spanning forests and critical bond percolations, and certain generalizations of gluing lemmas for bond percolation.

Based on joint work with Charles Newman and Vincent Tassion.

By Vladas Sidoravicius, NYU Shanghai/NYU

Continuity of phase transition is one among central questions in Statistical Physics. In my talk I will survey recent progress on this problem for Percolation, Ising, and two-dimensional Potts model.