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March 2327, 2015 YEP XII
Random walk in random environment
SUMMARY Random walk in random environment (RWRE) has originally been introduced in the early second half of the last century as a model for motions in disordered media, for the replication of DNA chains, and for phase transitions in alloys, among others. While the area has initially developed not too fast, in particular the last two decades have resulted in significant progress and a much deeper understanding of the topic. In fact, it has attracted significant mathematical attention and has undergone a major development, where, among others, results on limiting velocities, scaling limits, and large deviations have been established.
The YEP XII brings together many junior and some senior participants whose research interests are strongly related to RWRE. This mix of researchers aims at giving a special, open atmosphere which in the past has proven to be very fruitful for fostering discussions among the participants.
ORGANISERS
MINI COURSE SPEAKERS
INVITED SPEAKERS
Monday March 23
Tuesday March 24
Wednesday March 25
Thursday March 26
Friday
***************************************************************************************************************************************** ABSTRACTS Stein Bethuelsen Random Walk on Attractive SpinFlip Dynamics Consider random walk on the ddimensional lattice whose
transition probabilities are given by an underlying supercritical contact
process. This is a primal example of random walk in dynamic random environment,
which does not fall into the wellstudied conemixing class. We derive a
law of large number and derive bounds on large deviation probabilities for this
model. Noam Berger (mini course) The strongly ballistic phase for random walk in random environment The general case of random walk in random environment (RWRE) in dimensions larger than one is not well understood. However, if one makes some extra assumptions on the environment, then one can use several tools to get a clearer understanding of the behavior of the random walk. There are several different sets of assumptions, each enabling the use of a different set of tools, under which we can obtain interesting results. The purpose of this course is to focus on one such set of assumptions, known as strong ballisticity, and describe the progress in recent years in the understanding of the process under this set of assumptions. Oriane Blondel Random walks on the East model The East model is a onedimensional interacting particle
system with non attractive spinflip dynamics. In the physics literature, it is
a key example of a model with glassy features. Here we take this model as a
random environment and investigate the behaviour of two different random walks
whose jump rates depend on the current configuration. In particular, we are
interested in the relation between the dynamics of the East model and the
asymptotic speed and diffusion coefficient of the random walks. This contribution received funding from the INTEGER project Jiří Černı Ancestral lineages in spatial populations: Over the oriented random walk on oriented percolation cluster We study the long time behaviour of ancestral lineages in spatial population
models. These can be viewed as a random walk in a particular type of Markovian
random environment. The question is whether these lineages satisfy the CLT. David Croydon (mini course) Scaling limits of random walks on critical random trees and graphs This course will survey some recent work regarding the scaling limits of random walks on various models of random trees and other graphs at criticality, including critical GaltonWatson trees, the critical ErdosRenyi random graph and the uniform spanning tree in two dimensions. In each of these examples, the random structure considered exhibits some kind of interesting fractal behaviour in the limit. This leads to topological challenges in showing the associated random walks converge to a diffusion, and also gives rise to anomalous features for both the discrete and continuous processes (such as subGaussian transition density estimates). Dirk Erhard The parabolic Anderson model in a dynamic random environment: random conductances The parabolic Anderson model is a differential equation, which describes the
evolution of a field of particles performing independent nearest neighbor simple
random walks with binary branching: particles jump at rate $\kappa$, $\kappa >
0$, split into two and die at rates determined by the environment. We denote by
u(x,t) the mean number of particles at site x at time t conditioned on the
evolution of the environment. In this talk I will discuss a modified version of
this model. More precisely I introduce the case in which the random walks move
according to random walks among a field of random conductances ${\kappa(x,y)}_{x,y\in
\Z^d}$. For specific choices of the random environment I will discuss a link
between the exponential growth rate of the number of particles at the origin for
the original model to the one of the modified model. Onur Gün Branching random walks in random environments on hypercubes We study branching random walks on hypercubes with
spatially random branching rates. We explicitly describe the asymptotics of the
expected number of particles, which corresponds to mutationselection models on
random fitness landscapes, for various random environments, in particular, for
Gaussian landscapes as in the Random Energy Model (REM) of spin glasses. Brett Kolesnik The Cut Locus of the Brownian Map: Continuity and Stability The Brownian map is a random geodesic metric space which
is homeomorphic to the sphere, of Hausdorff dimension 4, and the scaling limit
of a wide variety of planar maps. In a sense, it is a purely random,
nondifferentiable surface. We strengthen the socalled confluence of geodesics
phenomenon observed at the root of the Brownian map, and with this, reveal
several properties of its rich geodesic structure. Júlia Komjáthy Fixed speed competition on the configuration model with infinite variance degrees In this talk we consider
competition of two spreading colors starting from single sources on the
configuration model with i.i.d.\ degrees following a powerlaw distribution with
exponent \( \tau\in (2,3) \). Sebastian Müller Rotorrouting on GaltonWatson trees A rotorrouter walk on a graph is a deterministic
process, in which each vertex is endowed with a rotor that points to one of the
neighbors. A particle located at some vertex first rotates the rotor in a
prescribed order, and then is routed to the neighbor the rotor is now pointing
at. In the talk we discuss the behavior of rotorrouter walks on GaltonWatson
trees and give a classification in recurrence and transience for transfinite
rotorrouter walks on these trees. Jan Nagel The Einstein relation in the random conductance model The Einstein relation says that for a motion under external force the derivative of the effective velocity at 0 (as a function of the strength of the force) is given by the diffusivityof the unperturbed motion. While successfully applied in both theoretical and experimental physics, rigorous proofs exist only for few models. We prove the Einstein relation for the random walk in Z^d, when the transition probabilities are determined by random conductancesof the edges, chosen independent and identically distributed and bounded away from 0 and infinity. The proof is based on Lebowitz and Rost's argument using
Girsanov's theorem to obtain an alternative description of the diffusivity and a
regeneration structure for the biased process, which is robust for small values
of the bias. Tal Orenshtein Excited random walk with periodic cookies We will discuss excited random walk on the integers in elliptic and identically piled environments with periodic cookies. This is a discrete time process on the integers defined
by parameters $p_1,...,p_M$ in $(0,1)$ for some positive integer $M$, where in
the $i$th visit to an integer $z$ the walker moves to $z+1$ with probability $p_{i
\mod M}$, and to $z−1$ with probability $1p_{i \mod M}$. The main result will
be presented is an explicit formula, in terms of $p_1,…,p_M$, for determining
recurrence, transience to the left, or transience to the right. As an
application one can easily construct transient walks even when the average drift
per period is zero. Alejandro Ramirez Quenched central limit theorem for random walk in ergodic spacetime environment We prove a quenched central limit theorem for random walk in a time dependent
random environment under a mild ergodicity assumption. PierreFrancois Rodriguez On levelset percolation for the Gaussian free field Ron Rosenthal Quenched invariance principle for simple random walk on clusters of correlated percolation models We derive a quenched invariance principle for simple random
walk on the unique infinite cluster for a general class of percolation models on
$\mathbb{Z}^d$, $d\geq2$. Michele Salvi The law of large numbers for the Variable Range Hopping model
The Variable Range Hopping
model is considered in Physics as an accurate representation of electrical
conduction in semiconductors. From the mathematical point of view, it represents
a prominent example of reversible longrange random walks on random point
processes, which generalize in several ways the classical random conductance
model on the lattice. We ask ourselves how an external field influences the
limiting velocity of the walk: So far, only very few models of biased random
walks with trapping mechanisms have been rigorously studied. A precise control
of the speed is also a fundamental step towards the proof of an Einstein
Relation for this model. Vladas Sidoravicius Random walks in dynamic environment  mutual interaction case During three lectures I will discuss a couple of examples of RWDRE, where environment is evolving in time and is affected by the walker. Few methods, including multi scale arguments and geometric Markov chain approach will be presented, as well as many open problems. Francois Siemenhaus Random walk driven by simple exclusion process We prove strong law of large numbers and an annealed
invariance principle for a random walk in a onedimensional dynamic random
environment evolving as the simple exclusion process with jump parameter
$\gamma$. Martin Slowik Random conductance model in a degenerate ergodic environment: Invariance principle and heat kernel behaviour Consider a continuous time random walk on the Euclidean
lattice $\mathbb{Z}^d$ in an environment of random conductances taking values in
$[0, \infty)$. The law of the environment is assumed to be ergodic with respect
to space shifts and satisfies some moment conditions. In this talk, I will
discuss recent results on a quenched invariance principle and local limit
theorem for this Markov process. In particular, I will explain how a quenched
invariance principle can be established in case the law of the random
conductances has an atom at zero. Renato Soares dos Santos Mass concentration in the parabolic Anderson model with doublyexponential tails We consider the solution of the heat equation with random potential on the ddimensional lattice with initial condition localised at the origin. The potential is supposed i.i.d. with upper tails close to doublyexponential. In this case, the solution is known to exhibit
intermittent behaviour, i.e., its mass is asymptotically concentrated on
relatively small "islands" that are wellseparated in space. The number of
islands needed is known to be a.s. asymptotically bounded by any small power of
time. We show that, with probability tending to one as time increases, most of
the mass of the solution is carried by a single island of bounded size. A
crucial ingredient in the proof is the recent characterization due to Biskup and
Koenig of the maxorder class of the principal eigenvalue of the Anderson
Hamiltonian in a growing box. Tobias Wassmer Aging of the Metropolis dynamics on the Random Energy Model Aging is one of the interesting features appearing in the longtime behavior of complex disordered systems such as spin glasses. The dynamics of these systems can be described by certain Markov chains in random environments. In the last two decades many works have proved aging for different spin glass models. These works however mostly lack some realism, as the dynamics considered are just a time change of simple random walk. In my talk I will present our recent progress in proving aging for the physically more realistic Metropolis dynamics. Joint work with Jiří Černı. Atilla Yilmaz Variational formulas and disorder regimes of random walks in random potentials I will start by providing three variational formulas for the quenched free
energy of a random walk in random potential (RWRP) when the underlying walk is
directed or undirected, the environment is stationary & ergodic, and the
potential is allowed to depend on the next step of the walk which covers RWRE.
Next, in the directed i.i.d. case, I will give two variational formulas for the
annealed free energy of RWRP. These five formulas are the same except that they
involve infima over different sets, and I will say a few words about how they
are derived. Then, I will present connections between the existence & uniqueness
of the minimizers of these variational formulas and the weak & strong disorder
regimes of RWRP. I will end with a conjecture regarding very strong disorder.
PRACTICAL INFORMATION ● VenueEurandom, Mathematics and Computer Science Dept, TU Eindhoven, Den Dolech 2, 5612 AZ EINDHOVEN, The Netherlands
Eurandom is located on the campus of
Eindhoven University of
Technology, in the
Metaforum building
(4th floor) (about
the building). The university is
located at 10 minutes walking distance from Eindhoven main railway station (take
the exit north side and walk towards the tall building on the right with the
sign TU/e).
● RegistrationRegistration is free, but compulsory for speakers and participants. Please follow the link: REGISTRATION PAGE
● AccommodationFor invited participants, we will take care of accommodation. Other attendees will have to make their own arrangements. We have a preferred hotel, which can be booked at special rates. Please email Patty Koorn for instructions on how to make use of this special offer. For other hotels around the university, please see: Hotels (please note: prices listed are "best available"). More hotel options can be found on the webpages of the Tourist Information Eindhoven, Postbus 7, 5600 AA Eindhoven.
● TravelFor those arriving by plane, there is a convenient direct train connection between Amsterdam Schiphol airport and Eindhoven. This trip will take about one and a half hour. For more detailed information, please consult the NS travel information pages or see Eurandom web page location. Many low cost carriers also fly to Eindhoven Airport. There is a bus connection to the Eindhoven central railway station from the airport. (Bus route number 401) For details on departure times consult http://www.9292ov.nl The University can be reached easily by car from the highways leading to Eindhoven (for details, see our route descriptions).
● Conference facilities : Conference room, Metaforum Building MF11&12The meetingroom is equipped with a data projector, an overhead projector, a projection screen and a blackboard. Please note that speakers and participants making an oral presentation are kindly requested to bring their own laptop or their presentation on a memory stick.
● Conference SecretariatUpon arrival, participants should register with the workshop officer, and collect their name badges. The workshop officer will be present for the duration of the conference, taking care of the administrative aspects and the daytoday running of the conference: registration, issuing certificates and receipts, etc.
● CancellationShould you need to cancel your participation, please contact Patty Koorn, the Workshop Officer. There is no registration fee, but should you need to cancel your participation after January 2, 2014, we will be obliged to charge a noshow fee of 30 euro.
● ContactMrs. Patty Koorn, Workshop Officer, Eurandom/TU Eindhoven, koorn@eurandom.tue.nl SPONSORSThe organisers acknowledge the financial support/sponsorship of:
Last updated
190615,
