George Tauchen : The Realized Laplace Transform of Volatility
- Probability ( 97 Views )We introduce a new measure constructed from high-frequency financial data which we call the Realized Laplace Transform of volatility. The statistic provides a nonparametric estimate for the empirical Laplace transform of the latent stochastic volatility process over a given interval of time. When a long span of data is used, i.e., under joint long-span and fill-in asymptotics, it is an estimate of the volatility Laplace transform. The asymptotic behavior of the statistic depends on the small scale behavior of the driving martingale. We derive the asymptotics both in the case when the latter is known and when it needs to be inferred from the data. When the underlying process is a jump-diffusion our statistic is robust to jumps and when the process is pure-jump it is robust to presence of less active jumps. We apply our results to simulated and real financial data.
Erika Berenice Roldan Roa : Asymptotic behavior of the homology of random polyominoes
- Probability ( 160 Views )In this talk we study the rate of growth of the expectation of the number of holes (the rank of the first homology group) in a polyomino with uniform and percolation distributions. We prove the existence of linear bounds for the expected number of holes of a polyomino with respect to both the uniform and percolation distributions. Furthermore, we exhibit particular constants for the upper and lower bounds in the uniform distribution case. This results can be extend, using the same techniques, to other polyforms and higher dimensions.
Gerandy Brito : Alons conjecture in random bipartite biregular graphs with applications.
- Probability ( 159 Views )This talk concerns to spectral gap in random regular graphs. We prove that almost all bipartite biregular graphs are almost Ramanujan by providing a tight upper bound for the second eigenvalue of its adjacency operator. The proof relies on a technique introduced recently by Massoullie, which we developed for random regular graphs. The same analysis allow us to recover hidden communities in random networks via spectral algorithms.
Nicolas Zygouras : Pinning-depinning transition in Random Polymers
- Probability ( 190 Views )Random Polymers are modeled as a one dimensional random walk (S_n), with excursion length distribution P(S_1 = n) = \phi(n)/n^\alpha, \alpha > 1 and \phi(n) a slowly varying function. The polymer gets a random reward whenever it visits or crosses an interface. The random rewards are realised as a sequence of i.i.d. variables (\omega_n). Depending on the relation between the mean value of the disorder \omega_n and the temperature, the polymer might prefer to stick to the interface (pinnings) or undergo a long excursion away from it (depinning). In this talk we will review some aspects of random polymer models. We will also discuss in more detail the pinning-depinning transition of the `Pinning' model and prove its annealed and quenched critical points are distinct. This is joint work with Ken Alexander.
Nayantara Bhatnagar : Subsequence Statistics in Random Mallows Permutations
- Probability ( 243 Views )The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied problem. Vershik-Kerov and Logan-Shepp first showed that asymptotically the typical length of the LIS is 2sqrt(n). This line of research culminated in the work of Baik-Deift-Johansson who related this length to the GUE Tracy-Widom distribution. We study the length of the LIS of random permutations drawn from the Mallows measure, introduced by Mallows in connection with ranking problems in statistics. We prove limit theorems for the LIS for different regimes of the parameter of the distribution. I will also describe some recent results on the longest common subsequence of independent Mallows permutations. Relevant background for the talk will be introduced as needed. Based on work with Ron Peled, Riddhi Basu and Ke Jin.
Maria Gordina : Gaussian type analysis on infinite-dimensional Heisenberg groups
- Probability ( 168 Views )This is a joint work with B.Driver. The groups in question are modeled on an abstract Wiener space. Then a group Brownian motion is defined, and its properties are studied in connection with the geometry of this group. The main results include quasi-invariance of the heat kernel measure, log Sobolev inequality (following a bound on the Ricci curvature), and the Taylor isomorphism to the corresponding Fock space. The latter is a version of the Ito-Wiener expansion in the non-commutative setting.
David Sivakoff : Random Site Subgraphs of the Hamming Torus
- Probability ( 143 Views )The critical threshold for the emergence of a giant component in the random site subgraph of a d-dimensional Hamming torus is given by the positive root of a polynomial. This value is distinct from the critical threshold for the random edge subgraph of the Hamming torus. The proof uses an intuitive application of multitype branching processes.
Robin PEMANTLE : Zeros of random analytic functions and their derivatives
- Probability ( 192 Views )I will discuss a series of results concerning the effect of the derivative operator on the locations of the zeros of a random analytic function. Two models are considered. In the first, the zeros are chosen IID from some measure on the complex plane. In the second, the zeros are chosen to be a Poisson point process on the real line. Repeated differentiation results in a nearly deterministic zero set.
Antonio Auffinger : The Parisi Formula: duality and equivalence of ensembles.
- Probability ( 177 Views )In 1979, G. Parisi predicted a variational formula for the thermodynamic limit of the free energy in the Sherrington-Kirkpatrick model and described the role played by its minimizer, called the Parisi measure. This remarkable formula was proven by Talagrand in 2006. In this talk I will explain a new representation of the Parisi functional that finally connects the temperature parameter and the Parisi measure as dual parameters. Based on joint-works with Wei-Kuo Chen.
Lisa Hartung : Extreme Level Sets of Branching Brownian Motion
- Probability ( 237 Views )Branching Brownian motion is a classical process in probability theory belonging to the class of Log-correlated random fields. We study the structure of extreme level sets of this process, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. We add to these results by finding the asymptotic size of extreme level sets and the typical height and shape of those clusters which carry such level sets. We also find the right tail decay of the distribution of the distance between the two highest particles. These results confirm two conjectures of Brunet and Derrida.(joint work with A. Cortines, O Louidor)
Scott Schmidler : Mixing times for non-stationary processes
- Probability ( 190 Views )Markov chain methods for Monte Carlo simulation of complex physical or statistical models often require significant tuning. Recent theoretical progress has renewed interest in "adaptive" Markov chain algorithms which learn from their sample history. However, these algorithms produce non-Markovian, time-inhomogeneous, irreversible stochastic processes, making rigorous analysis challenging. We show that lower bounds on the mixing times of these processes can be obtained using familiar ideas of hitting times and conductance from the theory of reversible Markov chains. The bounds obtained are sufficient to demonstrate slow mixing of several recently proposed algorithms including adaptive Metropolis kernels and the equi-energy sampler on some multimodal target distributions. These results provide the first non-trivial bounds on the mixing times of adaptive MCMC samplers, and suggest a way of classifying adaptive schemes that leads to new hybrid algorithms. Many open problems remain.
Krishna Athreya : Coalescence in Galton-Watson trees
- Probability ( 192 Views )Consider a Galton-Watson tree. Pick two individuals at random by simple random sampling from the nth generation and trace heir lines of descent back in time till they meet. Call that generation X_n. In this talk we will discuss the probability distribution of X_n and its limits for the four cases m <1, m=1, m greater than 1 but finite, and m infinite, where m is the mean offspring size.
Hao Shen : Stochastic PDEs and regularity structures
- Probability ( 195 Views )In this talk I will review the basic ideas of the regularity structure theory developed by Martin Hairer, as well as its applications to stochastic PDE problems. I will then discuss my joint work with Hairer on the sine-Gordon equation and central limit theorems for stochastic PDEs.
Anirban Basak : Ferromagnetic Ising Measures on Large Locally Tree-Like Graphs
- Probability ( 190 Views )We consider the ferromagnetic Ising model on a sequence of graphs $G_n$ converging locally weakly to a rooted random tree. Generalizing [Montanari, Mossel, and Sly (2012)], under an appropriate continuity" property, we show that the Ising measures on these graphs converge locally weakly to a measure, which is obtained by first picking a random tree, and then the symmetric mixture of Ising measures with + and - boundary conditions on that tree. Under the extra assumptions that $G_n$ are edge-expanders, we show that the local weak limit of the Ising measures conditioned on positive magnetization, is the Ising measure with + boundary condition on the limiting tree. The continuity" property holds except possibly for countably many choices of $\beta$, which for limiting trees of minimum degree at least three, are all within certain explicitly specified compact interval. We further show the edge-expander property for (most of) the configuration model graphs corresponding to limiting (multi-type) Galton Watson trees. This talk is based on a joint work with Amir Dembo.
F. Baudoin : Functional Inequalities: Probability and geometry in interaction
- Probability ( 190 Views )The talk will be an introduction to the world of functional inequalities with a geometric content. I will in particular focus on the family of log-Sobolev and Sobolev inequalities and show how these inequalities are closely connected to the geometry of the ambient space. I will mainly follow the Bakry-Ledoux approach to these inequalities which is is based on the notion of intrinsic curvature of a diffusion operator and at the end of the presentation will explain how these ideas have recently been used in sub-Riemannian geometry.
David Kaspar : Scalar conservation laws with Markov initial data
- Probability ( 153 Views )The inviscid Burgers' equation has the remarkable property that its dynamics preserve the class of spectrally negative L\'{e}vy initial data, as observed by Carraro and Duchon (statistical solutions) and Bertoin (entropy solutions). Further, the evolution of the L\'{e}vy measure admits a mean-field description, given by the Smoluchowski coagulation equation with additive kernel. In this talk we discuss ongoing efforts to generalize this result to scalar conservation laws, a special case where this is done, and a connection with integrable systems. Includes work with F. Rezakhanlou.
Brian Rider : Log-gases and Tracy-Widom laws
- Probability ( 190 Views )The now ubiquitous Tracy-Widom laws were first discovered in the context of the Gaussian Orthogonal, Unitary, and Symplectic Ensembles (G{O/U/S}E) of random matrix theory. The latter may be viewed as logarithmic gases with quadratic (Gaussian) potential at three special inverses temperatures (beta=1,2,4). A few years back, Jose Ramirez, Balint Virag, and I showed that that one obtains generalizations of the Tracy-Widom laws at all inverse temperatures (beta>0), though still for quadratic potentials. I'll explain how similar ideas (and considerably more labor) extends the result to general potential, general temperature log-gases. This is joint work with Manjunath Krishnapur and Balint Virag.
Stanislav Molchanov : On the random analytic functions
- Probability ( 217 Views )The talk will contain a review of several recent results on the analytic continuation of the random analytic functions. We will start from the classical theorem on the random Taylor series (going to Borel s school), but the main subject will be the random zeta function (which was introduced implicitly by Cramer) and its generalizations. We will show that true primes are not truly random , since zeta functions for the random pseudo-primes (in the spirit of Cramer) have no analytic continuation through the critical line Re (z) = 1/2.
Ivan Corwin : Brownian Gibbs line ensembles.
- Probability ( 156 Views )The Airy line ensemble arises in scaling limits of growth models, directed polymers, random matrix theory, tiling problems and non-intersecting line ensembles. This talk will mainly focus on the "non-intersecting Brownian Gibbs property" for this infinite ensemble of lines. Roughly speaking, the measure on lines is invariant under resampling a given curve on an interval according to a Brownian Bridge conditioned to not intersect the above of below labeled curves. This property leads to the proof of a number of previously conjectured results about the top line of this ensemble. We will also briefly touch on the KPZ line ensemble, which arises as the scaling limit of a diffusion defined by the Doob-h transform of the quantum Toda lattice Hamiltonian. The top labeled curve of this KPZ ensemble is the fixed time solution to the famous Kardar-Parisi-Zhang stochastic PDE. This line ensemble has a "softer" Brownian Gibbs property in which resampled Brownian Bridges may cross the lines above and below, but at exponential energetic cost. This is based on joint work with Alan Hammond.
Sayan Banerjee : Singular Reflected Diffusions
- Probability ( 346 Views )I will talk about some models coming from Physics and Queueing Theory that give rise to singular reflected processes in their diffusion limit. Such diffusions are characterized by non-elliptic generators (which are not even hypoelliptic) in the interior, and ergodicity arises from non-trivial interactions between the diffusion, drift and reflection. I will introduce a regenerative process approach which identifies renewal times in diffusion paths and analyzes excursions between successive renewal times. This provides a detailed description of the stationary distribution even when closed form expressions are unavailable. Based on joint works with Chris Burdzy, Brendan Brown, Mauricio Duarte and Debankur Mukherjee.
David Sivakoff : Nucleation scaling in jigsaw percolation
- Probability ( 190 Views )Jigsaw percolation is a nonlocal process that iteratively merges elements of a partition of the vertices in a deterministic puzzle graph according to the connectivity properties of a random collaboration graph. We assume the collaboration graph is an Erdos-Renyi graph with edge probability p, and investigate the probability that the puzzle graph is solved, that is, that the process eventually produces the partition {V}. In some generality, for puzzle graphs with N vertices of degrees about D, this probability is close to 1 or 0 depending on whether pD(log N) is large or small. We give more detailed results for the one dimensional cycle and two dimensional torus puzzle graphs, where in many instances we can prove sharp phase transitions.
Hendrik Weber : Convergence of the two-dimensional dynamic Ising-Kac model
- Probability ( 192 Views )The Ising-Kac model is a variant of the ferromagnetic Ising model in which each spin variable interacts with all spins in a neighbourhood of radius $\ga^{-1}$ for $\ga \ll1$ around its base point. We study the Glauber dynamics for this model on a discrete two-dimensional torus $\Z^2/ (2N+1)\Z^2$, for a system size $N \gg \ga^{-1}$ and for an inverse temperature close to the critical value of the mean field model. We show that the suitably rescaled coarse-grained spin field converges in distribution to the solution of a non-linear stochastic partial differential equation. This equation is the dynamic version of the $\Phi^4_2$ quantum field theory, which is formally given by a reaction diffusion equation driven by an additive space-time white noise. It is well-known that in two spatial dimensions, such equations are distribution valued and a \textit{Wick renormalisation} has to be performed in order to define the non-linear term. Formally, this renormalisation corresponds to adding an infinite mass term to the equation. We show that this need for renormalisation for the limiting equation is reflected in the discrete system by a shift of the critical temperature away from its mean field value. This is a joint work with J.C. Mourrat (Lyon).
Jonathan Mattingly : Noise induced stabilization of dynamical systems
- Probability ( 193 Views )We investigate an example of noise-induced stabilization in the plane that was also considered in (Gawedzki, Herzog, Wehr 2010) and (Birrell,Herzog, Wehr 2011). We show that despite the deterministic system not being globally stable, the addition of additive noise in the vertical direction leads to a unique invariant probability measure to which the system converges at a uniform, exponential rate. These facts are established primarily through the construction of a Lyapunov function which we generate as the solution to a sequence of Poisson equations. Unlike a number of other works, however, our Lyapunov function is constructed in a systematic way, and we present a meta-algorithm we hope will be applicable to other problems. We conclude by proving positivity properties of the transition density by using Malliavin calculus via some unusually explicit calculations. arXiv:1111.175v1 [math.PR]
Laurie Field : Relating variants of SLE using the Brownian loop measure
- Probability ( 192 Views )In this talk I will discuss a framework for transforming one variant of the SchrammLoewner evolution (SLE) into another. The main tool in this approach is the Brownian loop measure. A simple case is to relate the reversal of radial SLE to whole-plane SLE, which looks the same locally. Writing the formula one might naïvely expect fails, because the loop measure term is infinite. In joint work with Greg Lawler, we show that there is a finite normalized version of the loop measure term, and that with this change, the naïve formula relating the two SLEs becomes correct.
Ofer Zeitouni : Slowdown in Branching random walks and the inhomogeneous KPP equation
- Probability ( 196 Views )The classical result of Bramson gives a precise logarithmic correction to the speed of front propagation in one dimensional branching random walks and Brownian motions. I will discuss several variants of this model where the slowdown term is not classical.
Firas Rassoul-Agha : On the almost-sure invariance principle for random walk in random environment
- Probability ( 199 Views )Consider a crystal formed of two types of atoms placed at the nodes of the integer lattice. The type of each atom is chosen at random, but the crystal is statistically shift-invariant. Consider next an electron hopping from atom to atom. This electron performs a random walk on the integer lattice with randomly chosen transition probabilities (since the configuration seen by the electron is different at each lattice site). This process is highly non-Markovian, due to the interaction between the walk and the environment. We will present a martingale approach to proving the invariance principle (i.e. Gaussian fluctuations from the mean) for (irreversible) Markov chains and show how this can be transferred to a result for the above process (called random walk in random environment). This is joint work with Timo Sepp\"al\"ainen.
David Herzog : Supports of Degenerate Diffusion Processes: The Case of Polynomial Drift and Additive Noise
- Probability ( 211 Views )We discuss methods for computing supports of degenerate diffusion processes. We assume throughout that the diffusion satisfies a stochastic differential equation on Rd whose drift vector field X0 is ``polynomial'' and whose noise coefficients are constant. The case when each component of X0 is of odd degree is well understood. Hence we focus our efforts on X0 having at least one or more components of even degree. After developing methods to handle such cases, we shall apply them to specific examples, e.g. the Galerkin truncations of the Stochastic Navier-Stokes equation, to help establish ergodic properties of the resulting diffusion. One benefit to our approach is that, to prove such consequences, all we must do is compute certain Lie brackets.
Lea Popovic : Genealogy of Catalytic Populations
- Probability ( 211 Views )For neutral branching models of two types of populations there are three universality classes of behavior: independent branching, (one-sided) catalytic branching and mutually catalytic branching. Loss of independence in the two latter models generates many new features in the way that the populations evolve. In this talk I will focus on describing the genealogy of a catalytic branching diffusion. This is the many individual fast branching limit of an interacting branching particle model involving two populations, in which one population, the "catalyst", evolves autonomously according to a Galton-Watson process while the other population, the "reactant", evolves according to a branching dynamics that is dependent on the number of catalyst particles. We show that the sequence of suitably rescaled family forests for the catalyst and reactant populations converge in Gromov-Hausdorff topology to limiting real forests. We characterize their distribution via a reflecting diffusion and a collection of point-processes. We compare geometric properties and statistics of the catalytic branching forests with those of the "classical" (independent branching) forest. This is joint work with Andreas Greven and Anita Winter.