PLENARY TALKS
Krzysztof Burdzy (University of Washington)
Title: Archimedes' Principle for a Ball in Ideal Gas
Abstract: I will describe an approach to Archimedes' principle using classical mechanics mixed with some stochastic ideas. Joint work with Jacek Małecki.
Steve Evans (UC Berkeley)
Title: Markov chains associated with radix sort and PATRICIA, and their bridges
Abstract: Radix sort and PATRICIA (Practical Algorithm To Retrieve Information Coded In Alphanumeric) are two related algorithms that involve storing sequences of infinite binary strings as the leaves of rooted binary trees. Feeding these algorithms suitable random input sequences generates Markov chains of "growing" rooted binary trees. We characterize concretely what the infinite bridges of these transient chains are; that is, we determine all the ways it is possible to condition the chains to "behave at large times". Analytically, this is equivalent to determining the Doob-Martin boundaries of the chains.
Tom Hutchcroft (Caltech)
Title: Percolation on Finite Transitive Graphs
Abstract: In Bernoulli bond percolation, each edge of some countable graph is either deleted or retained independently at random with retention probability p; percolation theorists are interested in the geometry of the resulting random subgraph and how this geometry depends on the parameter p. Classically, probabilists have studied these problems primarily on infinite graphs (and a few very structured finite graphs like tori), while combinatorialists have been mostly interested in studying highly symmetric finite graphs like complete graphs or hypercubes. In this talk, I will overview this classical theory before outlining recent work developing a general theory of percolation on arbitrary finite transitive graphs (which are harder to understand than infinite graphs!). In particular, I will summarise our progress on the basic questions: When is there a phase transition for the emergence of a giant cluster? When is the giant cluster unique? How does this relate to percolation on infinite graphs? We will see that a surprisingly general and complete answer to each of these questions is possible starting only with the assumption of transitivity. Joint work with Philip Easo.
Kavita Ramanan (Brown University)
Title: Asymptotics of Interacting Particles on Sparse Random Graphs: from Typical to Atypical Behavior
Abstract: Interacting particle systems consist of collections of stochastically evolving particles, indexed by the vertices of a graph, where each particle’s transition rates depend only on the states of the neighboring vertices in the graph. Such systems arise as models in a wide variety of applications ranging from statistical physics to epidemiology. An important goal is the characterization of both typical and atypical (or large deviations) behavior of the empirical measures of these systems as the number of vertices goes to infinity. The setting when the underlying graph is the complete graph is classical and well understood. In this talk, we describe recent results in the complementary setting of sequences of uniformly sparse (random) graphs, which in particular elucidate the influence of graph topology on the dynamics. Along the way we establish a Sanov-type theorem for unimodular marked random graphs, which is of independent interest. This is based on joint works with I-H. Chen, A. Ganguly and S.Yasodharan.
Tianyi Zheng (UC San Diego)
Title: Liouville Property for Random Walks and Conformal Dimension
Abstract: Conformal dimension was introduced in the late 1980s by P. Pansu; it is a natural invariant in the study of the geometry of hyperbolic spaces and their boundaries. In this talk we will discuss how conformal geometry can be used to prove the Liouville property (that is, all bounded harmonic functions are constant) for random walks on iterated monodromy groups, when the limit set has Ahlfors-regular conformal dimension strictly less than 2. Joint work with N. Matte Bon and V. Nekrashevych.
OTHER INVITED TALKS
Pooja Agarwal (UC San Diego)
Title: An SPDE limit for stochastic networks with reneging
Abstract: We consider a stochastic network with N parallel servers in which impatient customers with i.i.d. service times and i.i.d. patience times enter service in the order of arrival, and abandon the queue if their waiting time exceeds the patience time. Such systems model phenomena in diverse fields, ranging from operations research to biochemical reaction networks. We represent the state of the system by a coupled pair of measure-valued processes and show that as the number of servers goes to infinity, the normalized state process converges to the unique solution of a non-standard stochastic partial differential equation, subject to a non-linear boundary condition. We also show that the infinite-dimensional limit process has a unique invariant measure. This is joint work with Kavita Ramanan and Haya Kaspi.
Abstract: Recent studies have shown that volatility exhibits a fractional behavior with a Hurst exponent H < 0.5, which challenges the typical perception of volatility. In this paper, we explore range-based proxies of the volatility process to confirm these findings on more widely available data and non-standard assets. Our analysis reveals that log volatility based on range-based estimators behaves like a fractional Brownian motion with H lower than 0.1 and that the rough fractional stochastic volatility model (RFSV) is a relevant volatility model. Additionally, we compare the prediction power of the RFSV model with that of the AR, HAR, and GARCH models and find that it outperforms these models in most cases. Furthermore, we revisit the finding that spot volatilities can be modeled by rough stochastic volatility-type dynamics and confirm this result using at-the-money options on the S&P500 index with a short maturity. Finally, we find that the Hurst parameter obtained from implied volatility-based approximations of spot volatility is of order 0.3, slightly larger than that usually obtained from historical data, likely due to a smoothing effect caused by the remaining time to maturity of the considered options.
Marianna Russkikh (Caltech)
Title: Aztec diamond and Lorentz-minimal surface
Abstract: In this talk we study a sequence of ’perfect t-embeddings’ of uniformly weighted Aztec diamond. We believe that these t-embeddings always exist and that they are good candidates to recover the complex structure of big bipartite planar graphs carrying a dimer model. We show that the associated origami maps of uniformly weighted Aztec diamond converge to a Lorentz-minimal surface S. We show that in the setup of the uniformly weighted Aztec diamond perfect t-embeddings can be used to prove convergence of gradients of height fluctuations to those of the Gaussian free field in the intrinsic metric of the surface S.