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**June 10, 2014**

Nick Travers, The Technion,

*Inversions and Longest Increasing
Subsequence for k-Card-Minimum Random
Permutations
*

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)

**May
27, 2014
**Patric Karl Glöde, Erlangen-Nürnberg and The
Technion,

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,

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,

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
*

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.

Mike Hochman, HUJI

An inverse theorem for the entropy of convolutions

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.

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)

Oleg Butkovsky, The Technion,

**March 18, 2014**

Adela Svejda, The Technion,

*"Clock processes on infinite graphs and aging in
Bouchaud's asymmetric trap model"
*

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.

Tom Meyerovitch, Ben Gurion,

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.

Eyal Neumann, The Technion

We study the solutions of the stochastic heat equation with spatially inhomogeneous white noise. This equation has the form \begin{equation} \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}. \end{equation} 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

(Joint work with Professor Leonid Mytnik)

January 7, 2014**
1030am Seminar 1:** Gennady Samorodnitsky, Cornell,

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)

Anna de Masi, Universiti di L'Aquila,

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.

Matthias Schulte, Karlsruhe,

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

Roman Kotecky, Charles Univ. and Warwick,

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).

Gidi Amir, Bar Ilan,

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**

Takashi Owada, The Technion,

*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).

For previous years see: Seminar Archive

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