Abstracts are added as they become available.
See the Schedule page for the schedule.

Extrema of two-dimensional discrete Gaussian free field and Liouville quantum gravity
Marek Biskup (University of California, Los Angeles)

Recent years have witnessed much progress in the understanding of the two-dimensional Discrete Gaussian Free Field (DGFF). In my talk I will discuss a specific aspect of this; namely, the asymptotic law of the extreme point process for the DGFF on lattice approximations of a bounded open set~$D$ in the complex plane with zero boundary conditions outside. It turns out that, for points arising from nearly-maximal local maxima, the limit process is Poisson with intensity that is the product of a random measure $Z^D(dx)$ in the spatial coordinate and the Gumbel intensity in the field coordinate. The conformal invariance of the continuum Gaussian Free Field manifests itself in the properties of the law of random measure $Z^D(dx)$. Indeed, this law obeys a canonical transformation rule under conformal maps of the domain~$D$ and inherits the Gibbs-Markov property of the DGFF. These permit us to link $Z^D(dx)$ to the measure representing the volume form of the critical two-dimensional Liou ville Quantum Gravity. The talk is based on joint work with Oren Louidor.

The Basel problem revisited: a quantum probabilistic proof
Zijian Diao (Ohio University)

Quantum computing is a revolutionary area of scientific research and technological development of the 21st century, where the principles of quantum mechanics are incorporated into the science of computation. By combining quantum Fourier transform and quantum amplitude amplification, two major ingredients of the celebrated Shor's factorization algorithm and Grover's search algorithm, we can carry out the simple counting procedure in a radically different fashion compared with its classical counterparts. The probabilistic behavior of this quantum counting algorithm exhibits intriguing connections with the classical Basel problem $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\dots=\frac{\pi^2}{6}$. In this talk, we will show a "quantum proof" of the original Basel problem as well as its variations.

Michael Damron (Indiana University, Bloomington)

In first-passage percolation (FPP), one places random non-negative weights on the edges of a graph and considers the induced weighted graph metric. Of particular interest is the case where the graph is $\mathbb Z^d$, the standard d-dimensional cubic lattice, and many of the questions involve a comparison between the asymptotics of the random metric and the standard Euclidean one. In this talk, I will give an introduction to some of the main lines of research in FPP, concentrating on my recent results with A. Auffinger (postdoc at Chicago), J. Hanson (postdoc Indiana) and P. Sosoe (grad student at Princeton). The topics will include the geometry and directional properties of geodesics, the set of possible limiting shapes for balls, and our most recent work, subdiffusive fluctuations of the metric. The last results build on work of Benjamini-Kalai-Schramm and Benaim-Rossignol.

Hölder continuity for the nonlinear stochastic heat equation with rough initial conditions
Le Chen (University of Utah)

We study space-time regularity of the solution of the nonlinear stochastic heat equation in one spatial dimension driven by space-time white noise, with a rough initial condition. This initial condition is a locally finite measure $\mu$ with, possibly, exponentially growing tails. We show how this regularity depends, in a neighbourhood of $t=0$, on the regularity of the initial condition. On compact sets in which $t>0$, the classical H\"older-continuity exponents $\frac{1}{4}-$ in time and $\frac{1}{2}-$ in space remain valid. However, on compact sets that include $t=0$, the H\"older continuity of the solution is $\left(\frac{\alpha}{2}\wedge \frac{1}{4}\right)-$ in time and $\left(\alpha\wedge \frac{1}{2}\right)-$ in space, provided $\mu$ is absolutely continuous with an $\alpha$-H\"older continuous density.

Transient, often anomalous and heterogeneous, diffusive transport through Nature's favorite barrier fluid: Mucus
M. Gregory Forest (University of North Carolina)

I will survey work with mathematical/statistical collaborators (with S. McKinley at the point) on experiments and data that seek to understand diffusive transport processes in the mucus barriers that protect every organ in the human body. Modern microscopy provides resolved path data for particles of varying size and chemical affinity to diverse mucus sources. The aims of these studies range from the ability to detect and distinguish health and disease progression from path data, to the ability to assess impacts of physical and drug therapies or vaccines, to the design of drug carrier particles to control arrival times of drugs through the mucus barrier to organic tissue. The translation of these applied aims into probabilistic/statistical problems will be presented, along with some progress to date. Our primary experimental collaborators are David Hill and Sam Lai at UNC, whereas our math team includes John Mellnik, Paula Vasquez, Natesh Pillai, Martin Lysy on pulmonary mucus and Alex Chen, Bill Shi, Simi Wang, Peter Mucha on cervicovaginal mucus.

A new method to generate uniform random variates on the unit spheric shell in $\mathbb R^d$
Barbara Gonzalez (Roosevelt University)

The generation of uniform random variates on the unit spheric shell is a key step in the simulation of many useful densities such as the multivariate normal and the multivariate Student's $t$. We present a novel algorithm to generate samples that are uniformly distributed on the unit spheric shell. Our method relies on recursively writing the joint density of the vector of random variables as the product of the marginal distribution of one component and the corresponding conditional distribution for the rest of the components, which after rescaling turns out to correspond to the density of another uniformly distributed random variable on a unit spheric shell of lower dimension. The results presented here can be especially useful to extend existing algorithms, like the Box-Muller method and the methods based on the properties of spacings.

Ornstein-Uhlenbeck processes for geophysical data analysis
Semere Habtemicael (North Dakota State University)

We propose the use of Gamma-Ornstein-Uhlenbeck processes and their modifications to analyze earthquake magnitude data. Such non-Gaussian Ornstein-Uhlenbeck processes offer the possibility of capturing important distributional deviations from Gaussianity and make the model flexible of dependence structures. Preliminary result shows that Gamma-Ornstein-Uhlenbeck process is a possible candidate for earthquake data modeling for some regions in the state of California. Development of a more sophisticated theory and more rigorous regression analysis is needed to extend the preliminary result to other earthquake-prone regions. Preliminary research correctly estimated a major earthquake date for five different regions in the state of California. A successful modeling will lead to estimation of a future major earthquake in any parts of the state of California.

Zeroes and critical points of random polynomials
Boris Hanin (Northwestern University)

The purpose of my talk is to present some results which prove that zeros and critical points of random polynomials in one complex variable come in pairs. I will explain how this pairing arises naturally by relating zeros and critical points to electrostatics on the Riemann sphere. Time permitting, I will also discuss generalizations of these results to zeros and critical points of random meromorphic functions on any closed Riemann surface.

Noise-induced stabilization of planar flows
David Herzog (Duke University)

With broad interests in mind, we discuss certain, explosive ODEs in the plane that become stable under the addition of noise. In each equation, the process by which stabilization occurs is intuitively clear: Noise diverts the solution away from any instabilities in the underlying ODE. However, in many cases, proving rigorously this phenomenon occurs has thus far been difficult and the current methods used to do so are rather ad hoc. Here we present a general, novel approach to showing stabilization by noise and apply it to these examples. We will see that the methods used streamline existing arguments as well as produce optimal results, in the sense that they allow us to understand well the asymptotic behavior of the equilibrium measure at infinity.

Immersed particle dynamics in fluctuating fluids with memory
Christel Hoheneggar (University of Utah)

Multibead passive microrheology aims at characterizing fluid properties via statistically measurable quantities (e.g. mean-square displacement, auto-correlation). Understanding how these material properties relate to biological quantities (e.g. exit time, first passage time through a layer) is of crucial importance. To correctly model the correlations due to the fluid's memory, it is necessary to include a thermally fluctuating force in the Stokes equations. We present such a model for an immersed particle passively advected by a fluctuating Maxwellian fluid. We describe the resulting stochastic partial differential equations and numerical method. Our approach can be applied to a Stokes fluid with memory created by a large suspension of active swimmers or to the diffusion of a particle in a crowded environment.

Threshold models for rainfall and convection: deterministic and stochastic triggers
Scott Hottovy (University of Wisconsin, Madison)

We study two types of convective triggers in models of single column water vapor for idealized single column dynamics. The models are idealizations of a general circulation model's convective parametrization trigger. In one model, we use a stochastic trigger that initiates strong convection at a random time after the water vapor has reached a critical value. In the other, the trigger is deterministic and occurs immediately when a fixed threshold is reached. With the stochastic trigger model it is difficult to analyze and compute statistics, while the deterministic trigger model has statistics which are easier to solve exactly. We show that the stochastic trigger model converges to the deterministic trigger model as the jump rate tends to infinity, and thus the deterministic model can be used as an approximation.

Brownian motion on Lie gorups: limits and fluctuations in large dimensions
Todd Kemp (University of California, San Diego)

Brownian motion on a Riemannian manifold is the diffusion process with generator equal to $\frac12$ the Laplacian on the manifold. The classical families of Lie groups, such as $\mathbb{U}_N$ and $\mathbb{GL}_N$, have canonical Riemannian metrics, and their Brownian motions have been well-understood for half a century. The question of what happens as $N\to\infty$ has only come into focus in the last two decades.
In this lecture, I will discuss the "strong law of large numbers" and the "central limit theorem" for these Brownian motions as $N\to\infty$. In the unitary case, one can make sense of these in terms of eigenvalues: the empirical spectral distribution forms a measure-valued process, which (as with the Hermitian case, given by the Dyson Brownian motion) converges almost surely to a measure-valued function, as shown by Biane in 1997. The fluctuations of this process are Gaussian, as proved by Lévy and Mäida in 2010. For non-normal matrices in $\mathbb{GL}_N$, the situation is much more difficult. By casting the question in a weak form, we can still understand the convergence of the Brownian motion as $N\to\infty$ as a "non-commutative measure" valued process. I will present some of my recent work describing this limit, solving a 20-year-old conjecture, and further discuss the fluctuations, which (as shown in recent joint work with Cébron) are also Gaussian.

Fractals at infinity and SPDEs
Kunwoo Kim (University of Utah)

In this talk, we first introduce the large scale Hausdorff dimension that was defined by Barlow and Taylor (1988). Using this notion of the dimension, we consider the behavior of various sets in R at infinity. In particular, we look at some SPDEs which exhibit high peaks and show how the Hausdorff dimensions change according to the heights of the peaks. This is joint work with Davar Khoshnevisan and Yimin Xiao.

Variational formula for limit shape of first-passage percolation
Arjun Krishnan (Courant Institute, New York University)

Consider first-passage percolation with positive, stationary-ergodic weights on the square lattice in $d$-dimensions. Let $T(x)$ be the first-passage time from the origin to $x$ in $\mathbb Z^d$. The convergence of $T([nx])/n$ to the time constant as $n$ tends to infinity is a consequence of the subadditive ergodic theorem. This convergence can be viewed as a problem of homogenization for a discrete Hamilton-Jacobi-Bellman (HJB) equation. By borrowing several tools from the continuum theory of stochastic homogenization for HJB equations, we will derive an exact variational formula (duality principle) for the time-constant. Under a symmetry assumption, we will use the variational formula to construct an explicit iteration that produces the limit shape.

Count of bi-modular hidden patterns under probabilistic dynamical systems

Consider a countable alphabet set $A$. A multi-modular hidden pattern is an $r$-tuple $(w_1,...,w_r)$, where each $w_i$ is a word over $A$ called a module. The hidden pattern is said to occur in a text $t$ when the later admits the decomposition $t=v_0w_1v_1...v_{r-1}w_rv_r$, for arbitrary words $v_i$ over $A$. It is known that multi-modular hidden patterns have asymptotically Normal frequencies in memoryless texts. About a decade ago it was conjectured that similar results should hold for more general and possibly non-Markovian models of texts. The technical difficulty for proving such results has been however the context-free nature of hidden patterns and the lack of logarithm- and exponential-type transformations to rewrite the product of non-commuting operators. In this talk, I will address a case study where we have successfully overpassed these difficulties and which may illuminate how to address more general cases. Specifically, I will show that the number of matches with a bi-modular pattern $(w_1,w_2)$ normalized by the number of matches with the pattern $w_1$, where $w_1$ and $w_2$ are different alphabet characters, is indeed asymptotically Normal when $(X_n)_{n\ge1}$ is produced by a holomorphic probabilistic dynamical source. This work is in collaboration with L. Lothe and was partially supported by the NSF DMS grant #0805950.

Memory and state dependent switching in biological diffusion
Scott McKinley (University of Florida)

The movement of nanoparticles in biological fluids like blood, mucus and cytoplasm is heavily influenced by physical and chemical interactions with a wide variety of environmental factors. Microscopy has revealed evidence that supports using the generalized Langevin equation and fractional Brownian motion to describe this motion in some settings while in other settings the movement is better described by state-dependent diffusion. In this talk I will give a brief survey of the physical mechanisms that give rise to these distinct behaviors as well as the statistical approaches that have been implemented to move toward rigorous model selection.

Laws Relating Runs and Steps in Gambler's Ruin

We calculate the conditional joint probability generating function of runs, $\mathbf{R}$, and steps, $\mathbf{L}$, given the height, $\mathbf{H}$, over a simple random walk excursion as a rational expression as follows. $E\{y^{\mathbf{R}}z^{\mathbf{L}}\,|\,\mathbf{H}\le N\}=C_Ny^2z^2\frac{q_N}{w_N}$, for a normalizing constant $C_N$. Here $q_N=q_N(x,u)$ and $w_N=w_N(x,u)$ have explicit closed form expressions as polynomials in $u:=y^2-1$ and $x:= z^2/4$, such that for $u=0$, $q_N(x,0)$ and $w_N(x,0)$ reduce to the Fibonacci-Chebychev polynomials in $x$ of index $N-1$ and $N$ respectively. This result extends the generating function representation of N.G. de Bruijn, D.E. Knuth, and S.O. Rice (1972), that corresponds to $u=0$. Consider the gambler's ruin $\mathbf{X}_j$ on the interval $[-N,N]$ started from $\mathbf{X}_0=0$, and denote by ${\cal R}_N$ the number of runs of the absolute value $|\mathbf{X}_j|$ of this process until the random epoch ${\cal L}_N$ of the so-called last visit by $\mathbf{X}_j$ to 0. Then, as $N\to\infty$, $N^{-2}{\cal R}_N$ converges in distribution to an absolutely continuous random variable with probability density $f(x)=2\sum_{\nu =1}^{\infty} e^{-\pi^2(2\nu-1)^2 x/4}=(\pi x)^{-1/2}\sum _{\nu = -\infty}^{\infty}(-1)^{\nu}e^{-\nu^2/x}, \mbox{ }x>0,$ and Laplace transform: $\int _{0}^{\infty}e^{-\lambda x}f(x)dx=\tanh(\sqrt{\lambda})/\sqrt{\lambda}$. In law, we find: $2\left(\lim_{N\to\infty}N^{-2}{\cal R}_N\right) =\lim_{N\to\infty}N^{-2}{\cal L}_N$. Further, denote by ${\cal R}'_N$ and ${\cal L}'_N$ the number of runs and steps respectively in the meander portion, that is subsequent to the last visit, of the gambler's ruin. Then, $N^{-1}\left(2{\cal R}'_N-{\cal L}'_N\right )$ converges in law as $N\to\infty$ to a density $s_2(x)=\frac{\pi}{4}\mbox{sech}^{2}(\pi x/2), \ -\infty < x < \infty.$

LAN property of some diffusion processes with jumps
Eulalia Nualart (Universitat Pompeu Fabra)

We consider an ergodic diffusion process with jumps driven by a Brownian motion and a Poisson random measure associated with a centered pure-jump Lévy process, whose drift coefficient depends on an unknown parameter. Supposing that the process is observed discretely at high frequency, we derive the local asymptotic normality (LAN) property. In order to obtain this result, Malliavin calculus and Girsanov's theorem are applied in order to write the log-likelihood ratio in terms of sums of conditional expectations, for which a central limit theorem for triangular arrays can be applied. Joint work with A. Kohatsu-Higa and N.K. Tran.

The growth model: Busemann functions, shape, geodesics, and other stories
Firas Rassoul-Agha (University of Utah)

I will consider the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles define solutions to variational formulas that characterize limit shapes and yield new results for Busemann functions, geodesics and the competition interface.

Obliquely reflected diffusions in non-smooth domains
Kavita Ramanan (Brown University)

Obliquely reflected diffusions in smooth domains are classical objects that have been well understood for half a century. Motivated by applications in a variety of fields ranging from mathematical physics to stochastic networks, a theory for obliquely reflected diffusions in piecewise smooth domains has also been developed over the last two decades. However, in domains with rough boundaries, even the definition of obliquely reflected diffusions is a challenge. We discuss an approach to constructing obliquely reflected diffusions in a large class of bounded, simply connected planar domains that may be extendable to higher dimensions. The class of processes we construct also includes certain processes with jumps like excursion-reflected Brownian motions, which have arisen in the study of SLE. This talk is based on works with Chris Burdzy, Zhenqing Chen, Weining Kang and Donald Marshall.

Update on the Beta ensembles
Brian Rider (Temple University)

The beta ensembles are one-parameter generalizations of the classical orthogonal and unitary invariant models of random matrix theory. They can also be viewed as one-dimensional caricatures of coulomb gases. In the last several years, various local spectral limit theorems for these ensembles have been characterized in terms of random differential operators. This includes, for instance, new descriptions of the ubiquitous Tracy-Widom laws. I'll summarize these developments (with a bias towards the picture at the spectral "edges") and sketch some open problems. This represents various joint projects with M. Krishnapur (IISC), J. Ramírez (Univ. Costa Rica), and B. Virág (Univ. Toronto).

Random walks in a sparse random environment
Youngsoo Seol (Texas State University, San Marcos)

We introduce random walks in a sparse random environment on the integer lattice Z and investigate such fundamental asymptotic property of this model as recurrence and transience criteria, the existence of the asymptotic speed and a phase transition for its value, limit theorems in both transient and recurrent regimes. The new model combines features of several existing models of random motion in random media and admits a transparent physics interpretation. More specifi cally, the random walk in a sparse random environment can be characterized as a perturbation of the simple random walk by a random potential which is induced by "rare impurities" randomly distributed over the integer lattice. The "impurities" in the media are rigorously defined as a marked point process on Z: The most interesting seems to be the critical (recurrent) case, where Sinai's scaling $(\log n)^2$ for the location of the random walk after n steps is generalized to basically $(\log n)^{\alpha}$ with $\alpha > 0$ being a parameter determined by the distribution of the distance between two successive impurities of the media.

The tug-of-war without noise and the infinity laplacian in a wedge
Robert Smits (New Mexico State University)

Consider the ending time of the tug-of-war without noise in a wedge. There is a critical angle for finiteness of its expectation when player I maximizes the distance to the boundary and player II minimizes the distance. There is also a critical angle such that for smaller angles, player II can fi nd a strategy where the expected ending time is finite, regardless of player I's strategy. For larger angles, for each strategy of player II, player I can fi nd a strategy making the expected ending time infi nite. Using connections with the inhomogeneous infinity Laplacian, we bound this critical angle.

Approximate counting with random walks
Florian Sobieczky (University of Denver)

The Delayed Random Walk on a disconnected graph is considered and the number of components is estimated. Criteria are given when this is possible in a time less than the conventional counting algorithms (such as the Hoshen-Kopelman method). Estimates of the number of open clusters per vertex of invariant percolation on an amenable graph are given.

Large deviations for the Erdös-Kac theorem
Lingjiong Zhu (University of Minnesota)

Erdös-Kac theorem is a celebrated result in number theory which says that the number of distinct prime factors of a uniformly chosen random integer satisfies a central limit theorem. In this talk, we discuss the large deviations for this problem. This is joint work with Behzad Mehrdad.