ColumbiaPrinceton Probability Day 2018Columbia University, Friday April 13, 2018. 


Senior Speakers


Schedule


Practical Information
The talks will take place at Columbia University, Hamilton Hall, Room 702 on April 13, 2018. Please fill the online registration form here if you are interested in coming. Directions to Columbia. For further information, please contact the organizers. 

Titles and Abstracts David Aldous Limits for processes over general networks Abstract: A typical nonelementary epidemic model consists of a model (with several parameters) for the contact network, and also a model (with several more parameters) for how infection spreads through the given network. Then assume a population size $n$ and $k_n$ initial infectives, where $k_n = \Omega(1)$ but $o(n)$. Intuitively there will be a phase transition: for some regions of parameter space there is a pandemic ($\Omega(n)$ eventually infected, with probability $\to 1$ as $n \to \infty$), and in other regions no pandemic (only $o(n)$ eventually infected, with probability $\to 1$ as $n \to \infty$), with a codimension $1$ "critical boundary" between these regions. This is true in analytically tractable models, but can one prove it for essentially arbitrary networks? We describe a very modest start to this kind of question. Based on joint work with Amir Dembo, Pablo Groisman, and Vladas Sidoravicius. Giovanni Peccati Limit theorems for Gaussian random waves Abstract: I will discuss several limit theorems characterising the highenergy fluctuations of geometric quantities (in particular, nodal lengths and counts) associated with Gaussian random waves  with specific emphasis on the planar, arithmetic and spherical cases. Our techniques are based on a pervasive use of WienerItô chaotic expansions, as applied to limit theorems and probabilistic approximations: in particular, we will show that the use of Wiener chaos fully explains some remarkable 'cancellation phenomenon' for variances of nodal quantities, first discovered by M. V. Berry in 2002 in the framework of planar random waves. Based on several joint works (some ongoing) with F. Dalmao, G. Dierickx, D. Marinucci, I. Nourdin, M. Rossi and I. Wigman. Kavita Ramanan Beyond MeanField Limits: Local Characterization of Interacting Processes on Sparse Graphs Abstract: Many applications can be modeled as a large system of homogeneous interacting particles on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical measure of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKeanVlasov or meanfield limit. In this talk, we focus on the complementary case of scaling limits of dynamics on sparse graphs, and obtain a novel characterization of the dynamics of a typical particle in the case of regular trees, and a class of GaltonWatson trees. The proofs rely on a certain Gibbs structure of the dynamics on the countably infinite graph $G$, which may be of independent interest. This is based on various joint works with Ankan Ganguly, Dan Lacker, Mitchell Wortsman and Ruoyu Wu. Konstantin Tikhomirov On the normal vector to a random hyperplane Abstract: Let H be a hyperplane spanned by i.i.d random vectors with independent coordinates with zero mean and unit variance. We study delocalization properties of a normal vector to H, extending and sharpening earlier results of Nguyen and Vu. We consider possible extensions of our method to delocalization of eigenvectors of nonHermitian random matrices. Joint work with Anna Lytova. Dan Romik A Pfaffian point process for Totally Symmetric Self Complementary Plane Partitions Abstract: Totally Symmetric Self Complementary Plane Partitions (TSSCPPs) can be encoded as a family of nonintersecting lattice paths having fixed initial points and variable endpoints. The endpoints of the paths associated with a uniformly random TSSCPP of given order therefore induce a random point process, which turns out to be a Pfaffian point process. I will discuss conjectural formulas for the entries of the correlation kernel of this process, and a more general "rationality phenomenon", which, if true, implies the existence of an interesting limiting process describing "infinite TSSCPPs", as well as conjectural probabilities for the occurrence of certain connectivity patterns in loop percolation (a.k.a. the dense O(1) loop model). Xin Sun Conformal embedding and percolation for triangulations Abstract: Liouville quantum gravity is a family of random surfaces which is conjectured to describe the scaling limit of conformally embedded random planar maps. We present a work in preparation proving this conjecture in the case of uniform triangulations under a discretization of the conformal embedding that is based on the conformal invariance of critical planar percolation. Joint work with N. Holden. 
