Short Programme on Probability and Statistical mechanics

Short Programme on Probability and Statistical mechanics

Content

Conference Agenda (updating)

Date:  Oct. 9 to Nov. 10, 2017
Register date & place: Oct. 8, 2017 at New Office Builing, AMSS, CAS
Venue: N202, New Office Building, AMSS, CAS
C1 is the identifier of mini-course given by Professor Auffinger. See the others in the list of mini-courses
S1 is the identifier of seminar talk given by Professor Auffinger. See the others in the list of seminars
 

Time Oct. 9 (Mon) Oct. 10 (Tues) Oct. 11 (Wed) Oct. 12 (Thur) Oct. 13 (Fri) Oct. 14 (Sat) Oct. 15 (Sun)
AM 09:30-10:30

C9 (I)

Hao Wu

 

C9 (II)

Hao Wu

 

C7 (I)

Zhonggen Su

C7 (II)

Zhonggen Su
C7 (III)

Zhonggen Su
10:30-11:30  
  Lunch
PM 14:30-15:30

C2 (I)

Dayue Chen


S3, Mufa Chen
 

C2 (II)

Dayue Chen

C6 (I)

Vladas Sidoravicius

S2, Gerard Ben Arous    
15:30-16:30 S14, Vladas Sidoravicius    
16:30-17:30            
  Dinner
  19:00-20:30              

 

Time Oct. 16 (Mon) Oct. 17 (Tues) Oct. 18 (Wed) Oct. 19 (Thur) Oct. 20 (Fri) Oct. 21 (Sat) Oct. 22 (Sun)
AM 09:30-10:30

C9 (III)

Hao Wu

 

C9 (IV)

Hao Wu

S5, Geoffrey Grimmett S6, Geoffrey Grimmett
S8, Zechun Hu
 
 
10:30-11:30     S10, Takashi Kumagai  
  Lunch
PM 14:30-15:30

C3 (I)

Andreas Kyprianou

S7, Yaozhong Hu

C3 (II)

Andreas Kyprianou

C6 (II)

Vladas Sidoravicius

C3 (III)

Andreas Kyprianou

   
15:30-16:30    
16:30-17:30       S9 (I), Daniel Kious S9 (II), Daniel Kious    
  Dinner
  19:00-20:30              

 

Time Oct. 23 (Mon) Oct. 24 (Tues) Oct. 25 (Wed) Oct. 26 (Thur) Oct. 27 (Fri) Oct. 28 (Sat) Oct. 29 (Sun)
AM 09:30-10:30

C9 (V)

Hao Wu

 

C9 (VI)

Hao Wu

 

C4 (I)

Leonardo Rolla

S17, Jiangang Ying  
10:30-11:30      
  Lunch
PM 14:30-15:30

S15, Jie Xiong

S11, Zenghu Li

 

C6 (III)

Vladas Sidoravicius

C4 (II)

Leonardo Rolla
   
15:30-16:30 S16, Lihu Xu  
16:30-17:30       S13, Alejandro Ramirez    
  Dinner
  19:00-20:30              

 

Time Oct. 30 (Mon) Oct. 31 (Tues) Nov. 1 (Wed) Nov. 2 (Thur) Nov. 3 (Fri) Nov. 4 (Sat) Nov. 5 (Sun)
AM 09:30-10:30   C5 (I)

Qi-Man Shao

C5 (III)

Qi-Man Shao

Workshop Agenda
Young Forum Agenda
 
10:30-11:30  
  Lunch
PM 14:30-15:30 S4, Shizan Fang C5 (II)

Qi-Man Shao
S18, Xicheng Zhang Workshop Agenda

Young Forum Agenda

 
15:30-16:30
16:30-17:30      
  Dinner
  19:00-20:30          

 

Time Nov. 6 (Mon) Nov. 7 (Tues) Nov. 8 (Wed) Nov. 9 (Thur) Nov. 10 (Fri)
AM 09:30-10:30  

C1 (I)

Antonio Auffinger

C1 (II)

Antonio Auffinger

C1 (III)

Antonio Auffinger

C8 (II)

Pierre Michel Tarres

10:30-11:30  
  Lunch
PM 14:30-15:30

 

S12, Kening Lu

S1, Antonio Auffinger

C8 (I)

Pierre Michel Tarres
C8 (III)

Pierre Michel Tarres
15:30-16:30  
16:30-17:30          
  Dinner
  19:00-20:30          


 

 Mini-Courses

 

C1 First-passage percolation: an introduction with many open problems
Speaker  Professor Antonio Auffinger, Northwestern University (USA)
Time  09:30-11:30 Nov. 7, 8 and 9 (3 lectures)
Place  N202
Abstract  This course is an introduction to one of the most classical models in probability theory, the model of first-passage percolation (FPP). During the lectures we will describe the main results of FPP, paying special attention to the recent burst of advances of the past 5 years. First, we will give self-contained proofs of seminal results obtained in the ’80s and ’90s on limit shapes and geodesics, while covering the state of the art of these questions. Second, aside from these classical results, we discuss recent perspectives and directions including focusing on the connection between Busemann functions and geodesics. Many interesting open problems will be presented.

 

C2 Interacting particle systems on trees
Speaker  Professor Dayue Chen, Peking University
Time  14:30-16:30, Oct. 9 and 11 (2 lectures)
Place  N202
Abstract  We shall review several models of interacting particle systems on trees: contact process, exclusion process, Ising model, etc. We will investigate the relation between the underlying space structure and the limiting behaviors of an interacting particle system. In particular we like to find out the difference of a particle system on a lattice and on a tree. We will list some problems to be studied.

 

C3 Exploration of $R^d$ by the isotropic alpha-stable process
Speaker  Professor Andreas Kyprianou, Bath University
Time  14:30-16:30 Oct. 16, 18 and 20 (3 lectures)
Place  N202
Abstract  In this mini-course we will review some very recent work on isotropic stable processes in high dimension. The recent theory of self-similar Markov and Markov additive processes gives us new insights into their trajectories. Combining this with classical methods, we revisit some old results, as well as offering new ones.

 

C4 Overview and Recent Progress in Absorbing-State Phase Transitions
Speaker  Professor Leonardo Rolla, NYU-Shanghai
Time  09:30-11:30, 14:30-16:30 Oct. 27 (2 lectures)
Place  N202
Abstract  *tentative only*

Modern statistical mechanics offers a large class of driven-dissipative stochastic systems that naturally evolve to a critical state, of which Activated Random Walks are perhaps the best example. The main pursuit in this field is to show universality of critical parameters, describe the critical behavior, the scaling relations and critical exponents of such systems, and the connection between driven-dissipative dynamics and conservative dynamics in infinite space.

This problem was stuck for more than a decade, then it saw significant partial progress about 9 years ago, and got stuck again. In this talk we will report on exciting progress made in the last 4 years, thanks to the contributions of Basu, Cabezas, Ganguly, Hoffman, Sidoravicius, Stauffer, Taggi, Teixeira, Tournier, Zindy, and myself. We will also discuss some of the several open problems.

 

C5 Basic Probability Inequalities and Stein's Method
Speaker  Professor Qi-Man Shao, Chinese University of Hong Kong
Time  09:30-11:30 Oct. 31 and Nov. 1, 14:30-16:30 Oct. 31 (3 lectures)
Place  N202
Abstract  In this short course we shall review some basic probability inequalities and Stein's method. Probability inequalities have played an important role in various proofs. Stein's method is a completely different method  from the classical Fourier method. It works for both independent and dependent random variables. It can also provide an accuracy of the approximation. The course will focus on the main idea of the method and recent development.

 

C6 Percolation - old problems and new challanges
Speaker  Professor Vladas Sidoravicius, NYU-Shanghai
Time  14:30-17:30 Oct. 12 and 26, 14:30-16:30 Oct. 19 (3 lectures)
Place  N202
Abstract  During this course I will introduce Bernoulli percolation model and other models related to Statistical Physics, such as Ising and Random Cluster models, and establish some basic properties, such as existence of the phase transition; new proof by Duminil-Copin and Tassion of the sharpness of the phase transition; the continuity of the phase transition in dimension two (Harris Theorem and Kesten Theorem). Next lectures will be devoted to renormalization techniques. In particular we will focus on Barsky-Grimmett-Newman method, and then discuss percolation on slabs, where we show its "well behavior" (Grimmett - Marstrand Theorem), and finally we will prove continuity of the phase transition on slabs. In the final lectures we will focus on the combinatorial/geometric methods to study phase transitions, in particular random currents representation and show continuity of the phase transition for the Ising model in any dimension. During this course I also will discuss many deep, mathematically challenging open problems.

 

C7 Determinantal Point Processes with Applications
Speaker  Professor Zhonggen Su, Zhejiang University
Time  09:30-11:30 Oct. 13, 14 and 15 (3 lectures)
Place  N202
Abstract  A determinantal point process is a random point process whose correlation functions can be expressed as a determinant of a kernel function. Determinantal point processes arise naturally in a number of apparently distinct probability models, and the focus is upon the study of kernel functions. In this short course, I shall first recall  some  basic concepts of point processes, correlations and provide some typical examples like Poisson processes,  GUE and Non-colliding random processes. And then I will turn to asymptotic behaviours for a few of common probability models using determinantal point processes. It includes the law of large numbers, the central limit theorems for the number of particles inside an interval, and Tracy-Widom type laws for the rightmost particles.

 

C8 Self-interacting random walks and statistical physics
Speaker  Professor Pierre Michel Tarres, NYU-Shanghai
Time  14:30-16:30 Nov. 9 and 10, 09:30-11:30 Nov. 10  (3 lectures)
Place  N202
Abstract  We will review various techniques for the study of self-interacting random walks, with a focus on the following topics: 1) Local time:  introduced by Ray-Knight for the study of Markov chains, the technique was first used in that context by Tóth in the 90s in a series of papers analysing a large class of self-interacting walks on the integer graph. In the non-exchangeable but self-repelling case, Tarrès, Tóth and Valkó  (resp. Horváth, Tóth and Vetö) were able to describe in 2012 that dynamics of the local time from the point of view of the particle for Brownian polymers (resp. self-avoiding walks), at the cost of losing information on the current position of the walk; see also Benaim, Ciotir and Gauthier (2015) in the compact case. 2) Coupling with Poisson Point Processes: initially proposed by Sellke (1994) for the study of strongly edge-reinforced random walks, the coupling was adapted to the study of vertex-reinforced random walk by Tarrès (2011), Basdevant, Schapira Singh (2014) (see also Kious (2016) for the so-called “stuck walks”)  3) Link with statistical physics, in particular the explicit correspondance between Edge-Reinforced Random Walk (ERRW), the supersymmetric hyperbolic sigma model (Sabot and Tarrès 2015),  the random Schrödinger operator (Sabot, Tarrès and Zeng 2016) and Dynkin's isomorhism (Sabot and Tarrès 2016).  

 

C9 Conformal invariance in critical 2D lattice models
Speaker  Professor Hao Wu, Tsinghua University
Time  09:30-11:30 Oct. 9, 11, 16, 18, 23, 25 (6 lectures)
Place  N202
Abstract 

Conformal invariance and critical phenomena in 2D statistical physics have been active areas of research in the last few decades, both in the mathematics and physics communities. In 1999, Oded Schramm introduced Schramm Loewner Evolution which provides a novel way to understand two-dimensional statistical lattice models. In this course, we will explain S. Smirnov’s work: the conformal invariance of the critical percolation and the conformal invariance of the critical Ising model.

Syllabus: 
Lecture 1: Bernoulli percolation
Lecture 2: Conformal invariance of Bernoulli percolation: Smirnov’s proof of Cardy’s formula
Lecture 3: Random-cluster model
Lecture 4: Conformal invariance of FK-Ising model
Lecture 5: Ising model
Lecture 6: Conformal invariance of Ising model

 

 Seminars (updating)

 

S1 The Sherrington-Kirkpatrick model
Speaker Professor Antonio Auffinger, Northwestern University
Time 14:30-16:30 Nov. 8
Place N202
Abstract In this talk, I will survey the main results on the archetype model of spin glasses in the past 10 years, the SK model. I will start with Parisi's prediction, explain the Parisi formula as proved by Talagrand, the ultrametricity theorem of Panchenko and end with a proof of full replica symmetry breaking at zero temperature. Talk is also based on results with Wei-Kuo Chen and Qiang Zeng.

 

S2 Complexity of random functions of many variables
Speaker Professor Gerard Ben Arous, Courant Institute
Time 14:30-15:30 Oct. 13
Place N202
Abstract Functions of many variables may be very complex. And optimizing them can be excedingly difficult or slow. Think for instance of the following simple question: How hard is it to find the minimum on the unit sphere of a cubic polynomial of many variables? If you chose the cubic polynomial randomly, it is very hard. Indeed it will have many local minima that will trap any algorithm trying to find the absolute minimum. First, I will adress the natural question: why does one care about this phenomenon? I will use examples in Data Science, and physics, where minimizing such functions is both important and hard. I will then explain briefly that one can describe and compute the complexity of functions of many variables, using the tools of Random Matrix Theory through a dictionary given by a classical formula of random geometry, the Kac-Rice formula.

 

S3 Mathematical Topics Motivated From Statistical Physics
Speaker Professor Mu-Fa Chen, Beijing Normal University
Time 14:30-16:30, Oct. 10
Place N202
Abstract The talk consists of two parts: (1) From equilibrium to non-equilibrium. A typical class of non-equilibrium particle systems --- reaction-diffusion processes; (2) The stability speed and the phase transitions. Spectral theory, the leading eigenvalue and so on. The first part is devoted to explore the mathematical foundation of statistical physics, the non-equilibrium systems in particular. The second part is to look for or to develop some mathematical tools for studying the phase transition phenomenon in statistical physics. The most part of the talk is informal.

 

S4 Lagrange's description of mechanics of fluids
Speaker Professor Shizan Fang, University of Bourgogne
Time 14:30-16:30, Oct. 30
Place N202
Abstract A survey for introductions.

 

S5 Counting self-avoiding walks
Speaker Professor Geoffrey Grimmett, Cambridge University
Time 09:30-10:30 Oct. 19
Place N202
Abstract The problem of self-avoiding walks (SAWs) arose in statistical mechanics in the 1940s, and has connections to probability, combinatorics, and the geometry of groups. The basic question is to count SAWs. The so-called 'connective constant' is the exponential growth rate of the number of n-step SAWs. We summarise joint work with Zhongyang Li concerned how the connective constant depends on the choice of graph. This work includes equalities and inequalities for connective constants, and a partial answer to the so-called ‘locality problem' for graphs and particularly Cayley graphs.

 

S6 The 1-2 model: dimers, polygons, and the complex Ising model
Speaker Professor Geoffrey Grimmett, Cambridge University
Time 09:30-10:30 Oct. 20
Place N202
Abstract The 1-2 model is a disordered interacting system in two dimensions which can be mapped to a number of well known processes including the dimer, Ising, and polygon systems of probability theory and mathematical physics. These connections will be explained in this lecture, and the exact solution will be summarised (joint work with Zhongyang Li).

 

S7 Some aspects of stochastic heat equations
Speaker Professor Yaozhong Hu, University of Alberta at Edmonton
Time 14:30-16:30 Oct. 17
Place N202
Abstract I will start with some basic concepts on stochastic heat equation.Then I will introduce and prove the Feynman-Kac formulas for the solution and for the moments of the solution. These formulas will be applied to obtain the sharp upper and lower bounds for the moments of the solution. Most of the talk are for noise determined by general Gaussian process applicable to fractional Brownian field with Hurst parameter greater than 1/2. I will also mention some recent work on the case Hurst parameter H<1/2.

 

S8 Hunt's Hypothesis (H) and Getoor's Conjecture
Speaker Professor Zechun Hu, Sichuan University
Time 09:30-11:30, Oct. 21
Place N202
Abstract This talk discusses Hunt's hypothesis (H) and Getoor's conjecture. It contains five parts. Firstly, I will talk about the background on Hunt's hypothesis (H). Secondly, I will recall the meaning of Hunt’s hypothesis (H) and its importance. Thirdly, I will introduce Getoor's conjecture and the existing results. Fourthly, I will introduce our results. Finally, I will mention some open problems. The talk is based on joints works with Wei Sun and Jing Zhang.

 

S9 The Once-reinforced random walk
Speaker Professor Daniel Kious, NYU Shanghai
Time 16:30-17:30, Oct. 19 and 20
Place N202
Abstract  The Once-reinforced random walk was introduced in the nineties as an a priori simplistic model of random walk with memory. Since then, only a few results have been proved, but very interesting features have been conjectured. During two one-hour talks, I will present in detail these conjectures about the recurrence-transience of the Once-reinforced random walk and I will give an overview of the current knowledge about this model. We will see different techniques which are useful in order to study the Once-reinforced reinforced random walks on ladders, or on trees. In particular, we will eventually focus on techniques developed in a recent joint work with Andrea Collevecchio and Vladas Sidoravicius.

 

S10 Convergence of random walks for trap models on disordered media
Speaker Professor Takashi Kumagai, Kyoto University
Time 10:30-11:30 Oct. 20
Place N202
Abstract In 2002, Fontes-Isopi-Newman introduced a diffusion (which is now called a FIN diffusion) as a scaling limit of the 1-dimensional Bouchaud trap model. It is a time change of 1-dimensional Brownian motion.

In this talk, we will consider more general setting and discuss convergence of random walks for trap models on disordered media. We first provide a general framework for studying such time changed processes and their discrete approximations in the case when the underlying stochastic process is strongly recurrent, in the sense that it can be described by a resistance form, as introduced by J. Kigami. We then apply the general theory to trap models on recurrent fractals and on random media such as the Erdős-Rényi random graph in the critical window. If time permits, we also discuss heat kernel estimates for the relevant time-changed processes.

This is a joint work with D. Croydon (Warwick) and B.M. Hambly (Oxford).

 

S11 Continuous-state branching processes in Levy environments
Speaker Professor Zenghu Li, Beijing Normal University
Time 14:30-16:30, Oct. 24
Place N202
Abstract A continuous-state branching processes in random environment (CBRE-process) is defined as the strong solution of a stochastic integral equation. The environment is determined by a Levy process with no jump less than -1. Characterizations of the quenched and annealed transition semigroups of the process can be given in terms of a backward stochastic integral equation. The process hits zero with strictly positive probability if and only if its branching mechanism satisfies Grey's condition. In that case, a characterization of the extinction probability is given using a random differential equation with singular terminal condition. The strong Feller property of the CBRE-process can be established by a coupling method. A criterion for the ergodicity of a subcricital CBRE-process with immigration can be given by an integral condition. This is a brief survey of the recent progresses in the subject.

 

S12 TBA
Speaker Professor Kening Lu, Brigham Young University
Time 14:30-16:30, Nov. 7
Place N202
AbstractTBA  

 

S13 Perturbative characterization of ballisticity of random walks in i.i.d. random environments
Speaker Professor Alejandro Ramirez, Pontificia Universidad Católica de Chile
Time 16:30-17:30 Oct. 27
Place N202
Abstract We provide a sufficient condition for ballisticity for random walks in i.i.d. random envi-ronments which are perturbations of the simple symmetric random walk in terms of the expectation and variance of the drift at a single site, extending and giving a sharper ver-sion of a result of Sznitman of 2004. Our theorem gives new examples of ballistic random walks in random environment satisfying the polynomial decay condition (P), but which do not satisfy Kalikow’s condition. This is a joint work with Santiago Saglietti.

 

S14 Mathematics of Multi-particle diffusion limited aggregation
Speaker Professor Vladas Sidoravicius, Courant and NYU-Shanghai
Time 15:30-16:30 Oct. 13
Place N202
Abstract In late seventies H. Rosenstock and C. Marquardt introduced the following stochastic aggregation model on $\mathbbZ^d$: Start with particles distributed according to the product Bernoulli measure with parameter $\?$. In addition, start with a static aggregate at the origin. Non-aggregated particles move as continuous-time simple random walks obeying the exclusion rule, whereas aggregated particles do not move. The aggregate grows by attaching particles to its surface whenever a particle attempts to jump onto it. This evolution is called Multi-particle Diffusion Limited Aggregation (MDLA). Its difficult to analyze structure inspired Witten and Sander to introduce in 1981 a "simplified version" of the model with only one particle moving at a step of procedure, which became well known as celebrated DLA model. MDLA model even in dimension 1 has highly nontrivial behavior. In its random walk version (instead of exclusion process) H. Kesten and V. Sidoravicius proved that if the original density of particles is smaller than one, then the aggregate is growing sublinearly, and more than ten years later A. Sly showed that for the density larger than one it advances linearly, establishing a remarkable phase transition. In my talk I will briefly review known results and will focus on the progress in dimensions $d\geq 2$. For random walk systems A. Sly argument imply linear growth of the far-most reaching arm of the aggregate. Our main result (joint with A. Stauffer) states that for the exclusion version of the process if $d > 1$ and the initial density of particles is large enough, then with positive probability the aggregate has linearly growing arms; that is, there exists a constant $c > 0$ so that at time $t$ the aggregate contains a point of distance at least $ct$ from the origin, for all $t$. In fact it obeys certain type of the shape theorem. The key conceptual element of our analysis is the introduction and study of a new growth process. Study of this process indicates that high density MDLA belongs to Kardar-Parisi-Zhang (KPZ) universality class. Very intriguing question is if in dimension $d \geq 2$ there exists similar phase transition as in dimension one, and how it affects geometric shapes of the aggregate. In other words is MDLA undergoing transition from DLA type growth (at low density) to KPZ type growth (for high density).

 

S15 BSDE and SPDE with Holder continuous coefficient and applications
Speaker Professor Jie Xiong, University of Macau
Time 14:30-16:30, Oct. 23
Place N202
Abstract The connection between backward stochastic differential equations (BSDE) and stochastic partial differential equations (SPDE) developed by Pardoux and Peng is extended to equation with Holder continuous coefficients. As applications, uniqueness of solutions for some SPDEs are obtained.

 

S16 Approximation of stable law by Stein’s method
Speaker Professor Lihu Xu, Universidade de Macau
Time 15:30-16:30, Oct. 25
Place N202
Abstract We will use Stein’s method to prove a general inequality which gives the convergence rate of stable law. Some examples will be studied with a comparison with the known results. The talk is based on the paper https://arxiv.org/abs/1709.00805

 

S17 On regular subspaces of Dirichlet forms
Speaker Professor Jiangang Ying, Fudan University
Time 09:30-11:30, Oct. 28
Place N202
Abstract This talk is a survey on regular Dirichlet subspace of a given Dirichlet form. We will talk about the background and recent progress of this problem. In particular, we will focus on the representation of symmetric Dirichlet forms on real line, and regular Dirichlet subspaces and extensions of 1-dim Brownian motion. More remained problems will be presented.

 

S18 Multidimensional singular stochastic differential equations
Speaker Professor Xicheng Zhang, Wuhan University
Time 14:30-16:30, Nov. 1
Place N202
Abstract In this talk I will report some recent progress about multidimensional singular stochastic differential equations. Here the singular SDE means that the coefficients are usually in some Sobolev spaces. I will address the basic method and some technical points. In particular, most of results are based on solving the associated PDE.