## Seminar on Probability and Stochastic Processes

#### To suggest a talk (yours or somebody else's) contact Nick Crawford, Oren Louidor or any member of our group.

To join the mailing list, contact Nick Crawford.

### Upcoming talks

June 10, 2014
Nick Travers, The Technion,
Inversions and Longest Increasing Subsequence for k-Card-Minimum Random Permutations

Abstract:
A random $n$-permutation can be generated with a deck of $n$ cards $D = {1,...,n}$ as follows. Draw a random card $C_1$ from the deck and remove it, then draw another random card $C_2$ from the remaining cards, and so forth until all $n$ cards have been removed. The permutation is $\sigma = (C_1,...,C_n)$ where $C_t$ is the card removed at time $t$. This permutation $\sigma$ is itself uniformly random as long as each random card $C_t$ is drawn uniformly from the remaining cards at time $t$.

We consider a variant of this simple procedure in which $k$ random cards are drawn from the deck independently at each step (with replacement) and the lowest numbered, or minimum, card is removed. Clearly, one expects the permutation $\sigma$ generated in this fashion to be more in order'' than a uniformly random permutation. That is, closer to the identity permutation $id = (1,2,3,...,n)$. We quantify this effect in terms of two natural measures of order: the number of inversions $I$ and the length of the longest increasing subsequence $L$. In particular, we characterize the scaling rate of $L$ as $n$ goes to infinity, both with fixed and growing $k$, and we prove a weak law of large numbers and central limit theorem for the number of inversions $I$, again both with fixed and growing $k$.

June 24, 2014

Ross Pinsky, The Technion, (tentative date)

### List of previous talks sorted backwards

May 27, 2014
Patric Karl Glöde, Erlangen-Nürnberg and The Technion,
Dynamics of Genealogical Trees for Autocatalytic Branching Processes

The subject of my talk are the dynamics of genealogical trees for autocatalytic branching populations. In such populations each individual has an infinitesimal death rate which depends on the total population size and, upon its death, produces a random number of offspring. I will consider finite as well as infinite populations. Formally, processes take values in the space of ultrametric measure spaces. While the discrete dynamics are constructed explicitly as piecewise deterministic Markov processes,  the limit dynamics for large populations are characterised by means of martingale problems. Key issues are proving well-posedness for the martingale problems and finding invariance principles linking finite and infinite populations. In fact, infinite populations arise as scaling limits of finite populations in the sense of weak convergence on path space with respect to the polar Gromov-weak topology. I will also show that there is a close relationship between the genealogies of infinite autocatalytic branching processes and the Fleming-Viot process. Moreover, I will present an abstract uniqueness result for martingale problems of, what I call, skew product form. This result is of importance for the processes discussed before but also applies to more general settings.

May 20, 2014
Roberto Fernandez, Utrecht,
Gibbs-non-Gibbs dynamical transitions.  A large-deviation paradigm

Abstract:
Ten years ago it was pointed out that a low-temperature Ising measure subjected to a high-temperature spin-flip evolution can become non-Gibbsian after a finite time. This was initially detected by relating evolving measures with static models known to exhibit non-Gibsianness.  A new paradigm is currently being explored, in which that Gibbs-non-Gibbs transitions correspond to bifurcations in the set of global minima of the large-deviation rate function for the measures trajectories conditioned on an observed end measure. I will present the main ideas behind this new approach to dynamic non-Gibbsianness and will report on rigorous results for mean-field and local mean-field models.

May 13, 2014
Yinon Spinka, TAU,
Phase structure of the loop O(n) model for large n

Abstract:
A loop configuration on the hexagonal (honeycomb) lattice is a finite subgraph of the lattice in which every vertex has degree 0 or 2, so that every connected component is isomorphic to a cycle. The loop O(n) model on the hexagonal lattice is a random loop configuration, where the probability of a loop configuration is proportional to x^(#edges) n^(#loops) and x,n>0 are parameters called the edge-weight and loop-weight. We show that for sufficiently large n, the probability that the origin is surrounded by a loop of length k decays exponentially in k. In this same region of parameters, we also show a phase transition from a disordered phase to an ordered phase.
No prior knowledge in statistical mechanics will be assumed. All notions will be explained.
Joint work with Hugo Duminil-Copin, Ron Peled and Wojciech Samotij.

April 29, 2014
Joseph Slawny, Virginia Tech and The Technion,
Self-dual spin systems: phase transitions and algebraic properties

Abstract:
I will start with a discussion of examples of classical translation invariant ferromagnetic spin systems. Then, using appropriate algebra, their low temperature phases will be described, self-dual systems will be introduced, their relation to binary error correcting codes (classical and quantum) will be explained, and most general translation invariant self-dual systems will be described. Time permitting, the problem of the location of zeros of the partition function of two-dimensional self-dual models and its relation to appropriate Weyl-Heisenberg group will be discussed.

April 22, 2014
Mike Hochman, HUJI
An inverse theorem for the entropy of convolutions

Abstract
A central theme in additive combinatorics is to understand how sets (and measures) grow under the sum (ersp. convolution) operation. The heuristic is that the result of these operations is substantially "larger" than the arguments unless there is an algebraic obstruction. The result I will describe in my talk is in this vein: it gives a statistical description of the structure of measures mu,nu on R^d with the property that mu*nu has only slightly more entropy than mu itself, at an appropriate scale. Roughly, small-scale pieces of mu,nu must be either essentially uniform or essentially atomic at intermediate scales. This is a variant of Freiman's theorem from additive combinatorics, more closely related to the work of Bourgain on the discretized ring conjecture. If time permits I will discuss applications.

April 8, 2014
Frank den Hollander, Leiden,
Random Walk in Dynamic Random Environment

Abstract:
In this talk I start with a brief outline of the main challenges for random walk in
dynamic random environment. After that I describe a one-dimensional example
where the dynamic random environment consists of a collection of particles
performing independent simple random walks in a Poisson equilibrium with
density $\rho \in (0,\infty)$. At each step the random walk moves to the right
with probability $p_{\circ}$ when it is on a vacant site and probability
$p_{\bullet}$ when it is on an occupied site. I show that when $p_\circ \in (0,1)$
and $p_\bullet \neq \frac12$, the position of the random walk satisfies a strong
law of large numbers, a functional central limit theorem and a large deviation
bound, provided $\rho$ is large enough. The proof is based on the construction
of approximate regeneration times, together with a multiscale renormalisation
argument.

Joint work with M. Hilario, R. dos Santos, V. Sidoravicius and A. Teixeira.

April 1, 2014
Wolfgang Loehr
Invariance principle for variable speed random walks on trees

Abstract:
Everyone knows that simple random walks on $\mathbb{Z}$ converge,
suitably rescaled, in path-space to Brownian motion on $\mathbb{R}$. This
has been generalised by Stone in 1963 to processes in natural scale'' on
$\mathbb{R}$, which are characterised by a speed measure $\nu$. We call
these processes speed-$\nu$ motions and generalise Stone's observation that they
depend continuously on $\nu$ to the case of trees.
More precisely, we show that whenever a locally compact $\mathbb{R}$-tree
$T_n$, together with a locally finite speed-measure $\nu_n$ on $T_n$,
converges in the Gromov-vague topology to a limiting $\mathbb{R}$-tree $T$
with measure $\nu$, the speed-$\nu_n$ motions converge to the speed-$\nu$
motion under a mild lower mass-bound assumption.
(joint work with Siva Athreya and Anita Winter)

March 25, 2014
Oleg Butkovsky, The Technion,
Convergence of Markov processes in the Wasserstein metric with applications

While convergence to stationary measure of finite dimensional Markov processes is quite well understood by now, less is known about convergence in the infinite-dimensional situation. In the first part of the talk we will present new sufficient conditions (in terms of the Lyapunov functions) for existence and uniqueness of an invariant measure of an infinite-dimensional Markov process. We will also provide explicit bounds on convergence rate. The second part of the talk will be devoted to applications of these theoretical results to various classes of Markov processes: stochastic functional differential equations, McKean-Vlasov equations, regime-switching diffusions, and SPDEs.

March 18, 2014
"Clock processes on infinite graphs and aging in Bouchaud's asymmetric trap model"

Abstract:
Clock processes of random dynamics in random environments have recently been at the center of attention in connection with the study of aging, which is a phenomenon that is believed to characterize the dynamical behavior of spin glasses. Based on a method for proving convergence of partial sum processes due to Durrett and Resnick, convergence criteria for clock processes were established for dynamics on finite graphs by Bovier and Gayrard. In this talk, we study dynamics that are defined on infinite graphs and present general convergence criteria for their clock processes. As an application we prove the existence of a normal aging regime in Bouchaud's asymmetric trap model on $\Z^d$ for all $d\geq 2$.
This talk is based on joint work with V. Gayrard.

March 10, 2014
Jesse Goodman, The Technion,
Detecting random graph degrees with traceroute sampling

Abstract:
In some large networks, notably the Internet, the actual network structure cannot be examined directly, and important network properties must be measured by inference. One way to do this is to use traceroute sampling: send out probes from a source vertex to every other vertex in the network, and examine the union of paths along which these probes travelled.

In mathematical terms, this operation corresponds to performing first passage percolation on the network.  A natural question is whether the partial network explored by this procedure accurately reflects the properties of the network as a whole or instead exhibits a bias.  In this talk we consider the configuration model with random edge weights, and I explain what is known about bias, and its absence, for the observed degrees.

January 21, 2014
Tom Meyerovitch, Ben Gurion,
Random Points on the metric polytope

Abstract:
We investigate a random metric space on n points constrained to have all distances smaller than 2, or in other words, we take a random point from the Lebesgue measure on the intersection of the so-called metric polytope with the cube [0,2]^(n(n-1)/2). We find that, to a good precision, the distances behave simply like i.i.d. numbers between 1 and 2.
Our proof uses an interesting mix of entropy methods and  the concept of "partial exchangeability".
Based on joint work with Gady Kozma Ron Peled and Wojciech Samtoij.

January 14, 2014
Eyal Neumann, The Technion
Pathwise Uniqueness of the Stochastic Heat Equations with Spatially Inhomogeneous White Noise

Abstract:
We study the solutions of the stochastic heat equation with spatially inhomogeneous white noise. This equation has the form $$\label{FracStoHeat} \frac{\partial}{\partial t} u(t,x) = \frac{1}{2}\Delta u(t,x) + \sigma(t,x,u(t,x))\dot{W} , \ \ t\geq0, \ \ x\in \mathbb{R}.$$ Here $\Delta$ denotes the Laplacian and $\sigma(t,x,u):\mathbb R_{+}\times\mathbb R^2\rightarrow \mathbb R$ is a continuous function with at most a linear growth in the $u$ variable. We assume that the noise $\dot{W}$ is a spatially inhomogeneous white noise on $\mathbb R_{+}\times\mathbb R$. When $\sigma(t,x,u)=\sqrt{u}$ such equations arise as scaling limits of critical branching particle systems which are known as catalytic super Brownian motion. In particular we prove pathwise uniqueness for solutions if $\sigma$ is Hölder continuous of index $\gamma>1-\frac{\eta}{2(\eta+1)}$ in $u$. Here $\eta\in(0,1)$ is a constant that defines the spatial regularity of the noise.

(Joint work with Professor Leonid Mytnik)

January 7, 2014
1030am Seminar 1:
General inverse problems for regular variation

Abstract:
Regular variation of distributional tails is known to be preserved by
various linear transformations of some random structures.
An inverse problem for regular
variation aims at understanding whether the regular variation of a
transformed random object is caused by regular variation of components
of the original random structure.  We derive results  of this type in the multivariate case
and in  situations where regular variation is
not restricted to one particular direction or quadrant.

(with E. Damek, T. Mikosch and J. Rosinski)

1130am Seminar 2: Alexander Drewitz, Columbia,
Asymptotics of the critical parameter for level set percolation of the Gaussian free field

Abstract:
We consider the Gaussian free field in $\textbf{Z}^d,$ $d \ge 3.$ It is known that there exists a non-trivial phase transition for
its level set percolation; i.e., there exists a critical parameter $h_*(d) \in [0,\infty)$
such that for $h < h_*(d)$ the excursion set above level $h$ does have a unique infinite connected component, whereas for
$h > h_*(d)$ it consists of finite connected components only.

We investigate the asymptotic behavior of $h_*(d)$ as $d \to \infty$ and give some ideas on the proof of this asymptotics.

(Joint work with P.-F. Rodriguez)

December 24, 2013
Anna de Masi,
Universiti di L'Aquila,
Stochastic particle systems and free boundary problems

Abstract:
The general topic of the talk is the hydrodynamic limit for  stochastic particle systems confined in a region. The hydrodynamic (macroscopic) equations have to be complemented with the boundary conditions: these are determined by the forces acting to keep the system confined in a bounded region.
The most studied case is when the boundary forces are due to reservoirs which fix the densities at the boundaries. If the boundary densities are non homogeneous, then the density gradient produces current that flow through the system according to Fick's  law.
The aim of this talk is instead to study the case when the region confining the system is determined by the state of the system itself. In continuum mechanics such situations are called free boundary problems and the prototipe is the Stefan problem were the system evolves according to the heat equation in a domain $\Om_t$ with Dirichlet boundary conditions and with the local speed of the boundary determined by the normal gradient of the solution.
I will present a one dimensional simple stochastic particle system were some of these issues can be analyzed in details (these results have been obtained in collaboration with G. Carinci, C. Giardin\a and E. Presutti).

December 17, 2013
Arnab Sen, Univ. of Minnesota,

Continuous spectra for sparse random graphs

Abstract:
The limiting spectral distributions of many sparse random graph models are known to contain atoms.  But do they also have some continuous part? In this talk, I will give affirmative answer to this question for several widely studied models of random graphs including Erdos-Renyi random graph G(n,c/n) with c > 1, random graphs with certain degree distributions and supercritical
bond percolation on Z^2. I will also present several open problems.

This is joint work with Charles Bordenave and Balint Virag.

December 10, 2013
Matthias Schulte, Karlsruhe,
Second order properties and central limit theorems for geometric functionals of Boolean models

Abstract: Boolean models are a fundamental topic of stochastic geometry and continuum percolation and have applications, for example, in physics, materials science and biology. The stationary Boolean model $Z$ is the random closed set that is formed by the union of compact and convex particles generated by a stationary Poisson process. The goal of this talk is to investigate random variables of the form $\psi(Z\cap W)$, where $\psi$ is an additive, translation invariant, and locally bounded functional and $W$ is a compact and convex observation window. Examples for $\psi$ are volume, surface area and Euler characteristic. For increasing observation windows asymptotic covariances are computed and univariate and multivariate central limit theorems are derived. In the important special case of intrinsic volumes of an isotropic Boolean model the covariance formulas can be further simplified. The proofs make use of the Fock space representation and the Malliavin-Stein method.

This is joint work with D. Hug and G. Last.

December 3, 2013
Roman Kotecky, Charles Univ. and Warwick,
Long range order for random colourings on planar lattices

Abstract:
We establish a phase transition (entropic long range order) for the uniform random perfect 3-colourings on a class of planar quasi-transitive graphs. The proof is based on an enhanced Peierls argument (which is of independent interest even for the Ising model for which it extends the range of temperatures with proven long range order) combined with an additional percolation argument. The motivation stemming from Potts antiferromagnet models will be explained.
Based on a joint work with Alan Sokal and Jan Swart.

November 26, 2013
Danny Vilenchik, The Weizmann Institute,
Chasing the k-colorability threshold

Abstract:
In this talk I will present a substantially improved lower bound on the $k$-colorability threshold of the random graph $G(n,m)$ with $n$ vertices and $m$ edges.
The new lower bound is 1.39 less than the $2k\ln k-\ln k$ first-moment upper bound (and 0.39 less than the $2k\ln k-\ln k-1$ physics conjecture).
By comparison, the best previous bounds left a gap of about 2+$\ln k$, unbounded in terms of the number of colors [Achlioptas, Naor: Annals of Mathematics 2005].
Furthermore, we prove that, in a precise sense, our lower bound marks the so-called condensation phase transition predicted on the basis of physics arguments.
Our proof technique is a novel approach to the second moment method, inspired by physics conjectures on the geometry of the set of $k$-colorings of the random graph.

Joint work with Amin Coja-Oghlan. Paper appeared in FOCS 2013.

November 19, 2013
Omri Sarig, Weizmann Institute, A random walk driven by an irrational rotation

Abstract:
The simple random walk can simulated dynamically as follows: Pick x randomly uniformly in the unit interval, and place a "walker" at the origin of $\mathbb Z$. Now  iterate the map $T(x)=2x mod 1$. Every time the orbit enters $[0,1/2)$, ask the"walker" to make a step to the left, and every time the orbit enters $[1/2,1)$ as the "walker" to make one step to the right. The result is the simple random walk.

The map $Tx=2x mod 1$ is highly "chaotic" (mixing,positive entropy,countable Lebesgue spectrum). What happens to the walk if we replace it by the highly "deterministic" (non-mixing, zero entropy, discrete spectrum)  irrational rotation $Tx=x+\alpha \mod 1$?.

I will describe some of the properties of the resulting random walk. (Joint with A.Avila, D.Dolgopyat, E. Duriev).

November 12, 2013
Gidi Amir, Bar Ilan,
The speed process of the Totally Asymmetric Zero-Range Processes.

Abstract:
Exclusion processes and Zero-range processes are two important examples of particle systems on Z that gathered a lot of attention in  statistical physics. In this talk I will focus on the totally asymmetric exclusion process (TASEP) and the constant-rate totally asymmetric zero-range process (TAZRP). I will define the two models and discuss  their basic properties such as their stationary measures. We will then introduce multi-type version of these processes (where particles of several types walk on Z according to the dynamics of the model, with lower type particles having priority over higher type ones).  In a paper with Angel and Valko, results on multi-type stationary measures for the TASEP and about the (random) speed of a second class particle in the "step" initial condition were used to define a speed process for the TASEP which turned out to have many interesting properties. I will briefly survey some results concerning that process, and then go on to describe a new speed process for the TAZRP. We will show how to generalize the  multi-type  "step" initial condition to the Zero-Range process and how to define a bijection between the models that allows to transfer some results in 2nd class particles to the TAZRP. We will then use these to define a speed process and analyze it. An important step is finding the stationary measures for the multi-type TAZRP, which will be analyzed as a system of queues in tandem.  We will also show how to use the speed process to make simple calculations of the speed of a 2nd class particle in the rarefacation fan for TAZRP under certain initial conditions.

The talk is based on joint work with O. Angel and B. Valko (TASEP) and current work in progress with P. Goncalves and J. Martin
No previous knowledge of the models involved will be assumed.

November 5, 2013
Maxima of long memory stationary symmetric $\alpha$-stable processes

Abstract:
We derive a functional limit theorem for the partial maxima process based on a long memory stationary $\alpha$-stable process. The length of memory in the stable process is parameterized by a certain ergodic theoretical parameter in an  integral representation of the process. The limiting process is no longer a classical extremal Fr\'echet process. It is a self-similar process with $\alpha$-Fr\'echet marginals, and it has the property of stationary max-increments, which we introduce in this paper. The functional limit theorem is established in the space $D[0,\infty)$ equipped with the Skorohod $M_1$-topology; in certain special cases the topology can be
strengthened to the Skorohod $J_1$-topology.

Based on joint work with G. Samoradnitsky.

October 29, 2013
Asaf Cohen, The Technion,
Diffusion Approximation in Bayesian Parameter Estimation

Abstract:
The starting point of the talk is whether one can use a posterior process of a diffusion process
(such as in the famous Zakai Equation model') in order to approximate a posterior process of
a discrete process which is relatively close to the mentioned diffusion process.

More specifically, I will talk a problem of Bayesian parameter estimation for a sequence of scaled counting processes
whose weak limit is a Brownian motion with an unknown drift. The main result is that the limit
of the posterior distribution processes is, in general, not equal to the posterior distribution of the mentioned
Brownian motion with the unknown drift. Surprisingly, it is equal to a posterior distribution process associated
with a different Brownian motion with an unknown drift. I will talk about the relation between the distributions
of these two Brownian motions with unknown drifts and will show that they can be arbitrary different.
The characterization of the limit of the posterior distribution processes is then applied to a stopping time problem.

October 22, 2013
Elliot Paquette, Weizmann Institute,
The spectral evolution of Erd\H{o}s-R\'{e}nyi random graphs near the
connectivity threshold and property (T) for random simplicial
complexes.

Abstract:
For $p = \Omega( n^{-1+\epsilon})$, standard random matrix theory
arguments can be used to show that all but the smallest eigenvalue of
the Laplacian of G(n,p) are $1+ \Theta((np)^{-0.5} ).$
Extensions by more specific graph methodology can bring this
down to p as small as $C \log n / n$.
At the connectivity threshold, this can no longer be true, as isolated
components contribute 0's to the spectrum.  Nonetheless, it makes sense
to consider the Laplacian of the remaining giant component of the
graph.  For these eigenvalues, we see that there is a phase transition
at $p = 0.5 \log n / n$.
We further consider the behavior of the codimension-2 links of a
Linial-Meshulam simplicial complex, which are marginally Bernoulli
random graphs.  For this object, we show a process version for all
links of the complex simultaneously and give some topological
consequences.

October 15, 2013
Oren Louidor, The Technion,
The thinned extremal process of the 2D discrete Gaussian Free Field.

Abstract:
We consider the discrete Gaussian Free Field in a square box of side N in $Z^2$ with zero
boundary conditions and study the joint law of its extreme values $(h)$ and their spatial positions $(x)$, properly centered and scaled. Restricting attention to extreme values which are also local maxima in a neighborhood of radius $r_N$, we show that when $N, r_N \to \infty$ with $r_N/N \to 0$, the joint law above converges weakly to a Poisson Point Process with intensity measure $Z(dx) e^{-\alpha h} dh$, where $\alpha = \sqrt{2\pi}$ and $Z(dx)$ is a random measure on $[0,1]^2$. In particular, this yields an integral representation for the law of the
absolute maximum, similar to that found in the context of Branching Brownian Motion. Time permitting, I will discuss various properties of the $Z$ measure, including connections with the derivative martingale associated with the continuum Gaussian Free Field. Joint work with Marek Biskup (UCLA).

***Webpage format shamelessly taken, like all good ideas, from Gady Kozma