December 12-16, 2016

"Guided Tour: Random Media"

Frank den Hollander

receives

Royal distinction


Photo credits: Rien Meulman



On Thursday 15 December, Frank den Hollander was appointed Knight in the Order of the Dutch Lion. The adornments were pinned on Den Hollander by Mayor Emile Jaensch of Oegstgeest, where Den Hollander resides. The ceremony occurred during the Eurandom workshop honouring the 60th birthday of Den Hollander.


Den Hollander received the distinction due to his career-long commitment to improve mathematics both nationally and internationally. The formalities took place in Eindhoven, where Den Hollander was second scientific director of the research institute Eurandom from 2000 until 2005. Den Hollander told Cursor – the independent news site of TU/e – that: “It was a very warm moment. Disseminating science is wonderful and you meet splendid people.”

The Royal distinction was presented on recommendation of the current scientific director of Eurandom, Remco van de Hofstad, and his predecessors Willem van Zwet and Onno Boxma. They describe Den Hollander as an eminent Dutch scientist who has done pioneering work internationally, in particular on the theory of large deviations. Furthermore, they praise Den Hollander as an “excellent director” who has “the respect needed to act as a leader and the vision needed to decide the proper course of action. In doing so, he takes mathematics and the mathematical community as the central focus.”

They also praise him for his warm character and energy, his positive and constructive attitude and his often-yearlong personal commitment to colleagues – from their first steps in research up until potentially their professorship.

Frank den Hollander, Professor of probability and statistical mechanics, is currently at Leiden University and one of the principal investigators of NETWORKS. He previously worked at TU/e and Radboud University Nijmegen.

 

SUMMARY

The aim of the proposed workshop is to bring together leading researchers working on various aspects of research involving random media and have them report on recent advances in their fields.
The intended audience has one additional feature in common; namely, they are all collaborators, former students or postdocs of Frank den Hollander, or their work has been heavily influenced by him. Frank den Hollander has been a central figure in research on random media over the last three decades and he managed to attract a lot of talented young people in it. As a result, the field is flourishing and many new research problems are being formulated and/or solved. We aim to run, first and foremost, a high-level scientific meeting.

In choosing the date we plan to hold our workshop close to Frank den Hollander’s 60th birthday on December 1, 2016 and use the occasion to celebrate his enormous contributions to the field.
 

ORGANISERS

INVITED SPEAKERS

PROGRAMME

MONDAY December 12

THEME: Polymers and Self-interacting Random Walks

TUESDAY December 13

THEME: Random Walks in (static and dynamic) Random Environment

WEDNESDAY December 14

THEME: Dynamical Gibbs-non-Gibbs Transitions and Path Large Deviations

THURSDAY December 15

THEME: Metastability
 

FRIDAY December 16

THEME: Parabolic Anderson Model and related topics

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ABSTRACTS

Luca Avena

Random walks in cooling random environments

We propose a model of a one-dimensional random walk in dynamic random environment that interpolates between two classical settings: (I) the random environment is resampled at every unit of time; (II) the random environment is sampled at time zero only (i.e. a static random environment). In our model the random environment is resampled along an increasing sequence of deterministic times. We consider the annealed version of the model, and look at three growth regimes for the resampling times: (R1) linear; (R2) polynomial; (R3) exponential. We prove laws of large numbers and central limit theorems. We list open problems and conjecture the presence of a crossover for the scaling behaviour in regimes (R2) and (R3).
(joint work with Frank den Hollander)

Ellen Baake

Solving the recombination equation

The recombination equation is a well-known dynamical system from mathematical population genetics, which describes the evolution of the genetic composition of a population that evolves under recombination. The genetic composition is described via a probability distribution (or measure) on a space of sequences of finite length; and recombination is the genetic mechanism in which two parent individuals are involved in creating the mixed sequence of their offspring during sexual reproduction. The model comes in a continuous-time and a discrete-time version; it can accommodate a variety of different mechanisms by which the genetic material of the offspring is partitioned across its parents. In all cases, the resulting equations are nonlinear and notoriously difficult to solve. Elucidating the underlying structure and finding solutions has been a challenge for nearly a century.
In this talk, we show how this equation can be solved, in two ways: forward in time, via a modern version of so-called Haldane linearisation; and backward in time via an associated stochastic fragmentation process. This is joint work with Michael Baake.
Reference: E. Baake, M. Baake, Haldane linearisation done right: Solving the recombination equation the easy way, Discr. Cont. Dyn. Syst. A 36 (2016), 6645 - 6656.

Michiel van den Berg

On Pólya’s inequality for torsional rigidity and first Dirichlet eigen-value

An inequality of Pólya asserts that for all open sets in Euclidean space with finite measure the product of torsional rigidity and first Dirichlet eigenvalue is bounded by its measure. We discuss the sharpness of this inequality and present some improvements for convex sets (joint work with Enzo Ferone, Carlo Nitsch and Cristina Trombetti). We also discuss bounds for the maximum of the torsion function in terms of the bottom of the spectrum of the Dirichlet Laplacian.

Matthias Birkner

Inhomogenous random walk pinning and conditional large deviations

The inhomogenous random walk pinning model is a Gibbs transform of a random walk where intersections with another walk are rewarded/penalized with random weights. It is connected to various polymer models, in particular to directed polymers in random environment, via partial annealing, and undergoes a localization/delocalization transition in high dimensions. We discuss how partial annealing can be formulated in terms of conditional large deviation principles for the empirical process obtained by cutting an i.i.d. sequence along a renewal process and what this implies for the comparison of critical points.

Erwin Bolthausen

On two-dimensional random walks in random environments

We report about work in progress with Erich Baur and Ofer Zeitouni. For the standard model of random walks in random environments, dimension two is the critical dimension. This means that in leading order, the disorder stays constant after a renormalization procedure. However, fine analysis of the corrections to the leading order reveal that the disorder is indeed contracting. We give a detailed explanation of this effect.

Anton Bovier

Metastability according to Frank

I will give a review on the developments in the theory of metastability mostly from the perspective of the contributions made by Frank, starting with his papers from
around 2000 on Kawasaki dynamics.

Francesco Caravenna

Universality in marginally relevant disordered systems

We consider disordered systems of directed polymer type, for which disorder is so-called marginally relevant. These include the usual directed polymer in random environment in dimension (2+1), as well as the disordered pinning model with tail exponent 1/2. We show that in a suitable weak disorder and continuum limit, the partition functions of such models converge to a universal limit. Connections with the two-dimensional Stochastic Heat Equation will be discussed.
(joint works with Rongfeng Sun and Nikos Zygouras)

Jean-Dominique Deuschel

Invariance principle for random walks with time-dependent ergodic degenerate weights

We study a continuous-time random walk, X, on Zd in an environment of dynamic random conductances taking values in (0,∞). We assume that the law of the conductances is ergodic with respect to space-time shifts. We prove a quenched invariance principle for the Markov process X under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by Moser's iteration scheme.
(joint work with S. Andres, A. Chiarini, and M. Slowik)

Roberto Fernandez

Gibbs-non-Gibbs dynamical transitions: The large-deviation paradigm and how we got there

The large-deviation paradigm has led to an alternative interpretation of Gibbs--non-Gibbs transitions in terms of the number of optimal conditioned trajectories of evolving measures. This offers a new insight into the phenomenon that translates into the detection of further instances of these transitions. This talk will review the the definition and first examples of these transitions, contrast old and new approaches and present benchmark examples of the large-deviation paradigm.

Cristian Giardina

Dynamical properties of the inclusion process.

The inclusion process is an interacting system made of particles that move on a graph G according to a reversible random walk kernel and, furthermore, they have a preference to accumulate locally on a few vertices. It has been introduced as a model of "heat conduction" enjoying duality properties of algebraic nature; it is also related to the Moran model of population genetics.
I will discuss the dynamical properties of the inclusion process in the condensation regime. There, in the appropriate scaling limit, the condensate is restricted to a subset of vertices G^* (the maxima of the random walk reversible measure) and the following metastability scenario with possibly multiple time-scales emerges. If the restriction of the random walk kernel to G^* is irreducible, then the system has only one time-scale (that we determine together with the law of the limiting process). However, if such restriction is reducible into several connected components, then there exist up to three time-scales: a first time-scale over which the system moves within connected components; a second time-scale to see the jumps between components that are at graph distance equal to two; a third (even longer) time-scale for the jumps between components that are at graph distance larger than two.
(joint work with Alessandra Bianchi and Sander Dommers)

Jesse Goodman

Long paths in first passage percolation on the complete graph

In a connected graph with random positive edge weights, pairs of vertices can be joined to obtain an a.s. unique path of minimal total weight. It is natural to ask about the typical total weight of such optimal paths, and about the number of edges they contain. To this end we consider the first passage percolation exploration process, which tracks the flow of fluid traveling across edges at unit speed and therefore discovers optimal paths in order of weight. On the complete graph, adding exponential edge weights results in optimal paths with logarithmically many edges - the same "small world" path lengths that are typical of many random graphs. However, by changing the edge weight distribution, we can obtain paths that are asymptotically longer than logarithmic. This talk will explain how tail properties of the edge weight distribution can be translated quite precisely into scaling properties of optimal paths.

Andreas Greven

Populations, genealogies and fluctuating environments

We consider spatial population models and allow the coefficients of the dynamic to be random themselves, an old topic in probability. We come to a new point here. Typically for such evolutions we can define a genealogy even for models defined as solutions to SSDE. This we explain. Consecutively we point out that spatially homogeneous models can often be viewed as dynamic in a randomly fluctuating medium leading to genealogies evolving in randomly fluctuating medium, the medium given via the process of local population sizes. This is demonstrated with the example of interacting logistic Feller diffusions.

Markus Heydenreich

Random walk driven by interacting particle systems

Some 10 years ago, Frank (dH) & Frank (Redig) started a research program investigating random walk in dynamic random environment involving Luca Avena, Florian Völlering, Renato dos Santos and many others. The object of study are random walks whose transition rates depend on an underlying random medium. The key point is then to let the random medium evolve in time. This can be viewed as an interpolation between classical RWRE models (where the medium is fixed in time) and the homogeneous model. The challenge is to understand the impact of the dynamic of the medium on the random walk characteristics.
I shall review some of this development, and then present a result obtained with Stein Bethuelsen about random walk driven by the contact process.

Wolfgang König

The spatially discrete parabolic Anderson model with time-dependent potential

I will review Frank's works since 2006, jointly with Jürgen, Gregory, and Dirk, on the long-time behaviour of the parabolic Anderson model on the $d$-dimensional lattice with time-dependent random potential.
That is, I will consider a random walk on the lattice that runs through a timely varying random potential (assumed space-time ergodic and sufficiently integrable) and carries a mass that is increased, respectively decreased, on the way, depending on the particular value of the potential. Main examples of potentials are given by fields of random walkers with or without interaction (e.g., an exclusion process or just independent walkers or just one single walker), where the potential is put equal to a parameter times the number of these walkers at a given place at a given time.  If this parameter is positive, these walkers can be interpreted as moving catalysts supporting a chemical reaction at the site of the reactant, the walker mentioned at the beginning.
The main question is about the large-time logarithmic asymptotics of the expected total mass of the reactant, in particular its behaviour as a function of the interaction strength parameter and of the diffusivity parameter of the reactant, in particular if this decreases to zero respectively increases to infinity. Here interesting comparisons with the almost-sure logarithmic asymptotics appear, which give rise to a deeper understanding of the entire particle system.

Roman Kotecky

Renormalization group in action

Renormalization group transformations, when intrepreted as a flow in the space of Hamiltonians, are often leading to non-Gibbs states. I will argue that properly formulated transformation on an extended space leads to useful applications. One example is the investigation of the strong convexity of the surface tension for random interfaces with nonconvex potentials.

Christiaen Maes

Path-large deviations approach to GENERIC

The talk reports on work in progress with Richard Kraaij, Alexandre Lazarescu and Mark Peletier.  The central question concerns the structure of hydrodynamic and kinetic equations in describing the relaxation to
equilibrium.  GENERIC stands for an extension of gradient flow, adding a Hamiltonian part to the dissipative term in the evolution equation.  To understand its appearance from the point of view of path large deviations we derive a new fluctuation symmetry.

Gregory Maillard

Parabolic Anderson model in a dynamic random environment: random conductances

We consider a version of the Parabolic Anderson model where the underlying random walk is driven by random conductances and investigate the effect on the Lyapunov exponents. We will show that the annealed Lyapunov exponents are controlled by pockets where the conductances are close to the value that maximises the growth in the homogeneous setting (constant conductances). In contrast, the quenched Lyapunov exponent is controlled by a mixture of pockets where the conductances are nearly constant.
(joint work with Frank den Hollander and Dirk Erhard)

Julián Martinez

Variational description of Gibbs-non-Gibbs dynamical transitions for spin-flip systems with a Kac-type interaction

We discuss the concept of Gibbs/ non-Gibbs measure in the mean field context and its extension to a local-mean field model, and the emergence of dynamical Gibbs-non-Gibbs transitions under independent spin-flip ("infinite-temperature") dynamics. We show that these dynamical transitions are equivalent to bifurcations in the set of global minima of the large-deviation rate function describing optimal conditioned trajectories of the empirical density. Possible bifurcation scenarios are fully determined in the mean field case, yielding a full characterization of passages from Gibbs to non-Gibbs -and vice versa- with sharp transition times.
(joint work with Roberto Fernández and Frank den Hollander)

Francesca Nardi

Competing Metastable States for general rare transition dynamics: model independent results, applications to Blume-Capel model and Probabilistic Cellular Automata

Metastability is a physical phenomenon ubiquitous in first order phase transitions. A fruitful mathematical way to approach this phenomenon is the study of rare transitions Markov chains. For Metropolis chains associated with statistical mechanics systems, this phenomenon has been described in an elegant way in terms of the energy landscape associated to the Hamiltonian of the system. We provide a similar description in the general rare transitions setup. Moreover, the study of systems with multiple (not necessarily degenerate) metastable states presents subtle difficulties from the mathematical point of view related to the variational problem that has to be solved in these cases. We give sufficient conditions to identify multiple metastable states. Since this analysis typically involves non-trivial technical issues, we give different conditions that can be chosen appropriately depending on the specific model under study. We show how these results can be used to attack the problem of multiple metastable states. Beside their theoretical content, our general results are a useful tool to approach metastability for the Blume–Capel model for a particular choice of the parameters for which the model has two multiple not degenerate in energy metastable states and for non-Metropolis systems such as Probabilistic Cellular Automata. In particular we estimate in probability and in expectation the time for the transition from any of the metastable states to the stable and we identify the set of critical configurations that represent the minimal gate for the transition.

Nicolas Pétrélis

Collapse transition of the 2-dimensional self-interacting prudent walk

In this talk we will consider a model of self interacting prudent walk in dimension 2.
The paths considered are self-avoiding and satisfy an additional constraint, i.e., they can not take a step in the direction of a site already visited. The uniform measure on the set of prudent paths is perturbed with the help of an Hamiltonian that rewards self-touchings.
We will show that such a model undergoes a collapse transition between an extended phase and a collapsed phase inside wich the free energy is linear. As an intermediate step of the proof we will show that te exponential growth rate of those two-sided prudent paths equals that of generic prudent paths, answering an open question raised in several papers before.
(joint work with Niccolò Torri)

Elisabetta Scoppola

Irreversible dynamics at low temperature: metastability and convergence to equilibrium

Some results on irreversible dynamics are presented pointing out differences between reversible and irreversible dynamics.

Gordon Slade

Critical exponents for long-range O(n) models below the upper critical dimension

We consider the critical behaviour of long-range O(n) models for n greater than or equal to 0.  For n=1,2,3,..., these are n-component phi^4 spin models.  For n=0, it is the weakly self-avoiding walk.  For all n=0,1,2,..., we prove existence of critical exponents for the susceptibility and the specific heat, slightly below the upper critical dimension. This is a rigorous version of the epsilon expansion in physics. The proof is based on a rigorous renormalisation group method developed in previous work with Bauerschmidt and Brydges.

Jan Swart

Eigenmeasures and sharpness of the phase transition for the contact process

It is well known that for contact processes, extinction is exponentially fast in the whole subcritical regime. In this talk, I will present a new and surprisingly short proof of this fact that is based on eigenmeasures, which are possibly infinite measures on the set of nonempty configurations that are preserved under the dynamics up to a time-dependent exponential factor. In particular, in the subcritical regime, there is a one-to-one correspondence between translation-invariant eigenmeasures and quasi-invariant laws of the process modulo translations.

Rongfeng Sun

Subdiffusivity of a random walk among a Poisson system of moving traps on Z

We consider a random walk among a Poisson system of moving traps on Z. Previously, the quenched and annealed survival probabilities of this random walk have been investigated. Here we study the path of the random walk conditioned on survival up to time t in the annealed case and show that it is subdiffusive. As a by-product, we obtain an upper bound on the number of so-called thin points of a one-dimensional random walk, as well as a bound on the total volume of the holes in the random walk's range. Based on joint work with S. Athreya and A. Drewitz (to appear in Mathematical Physics, Analysis and Geometry).

Jan Swart

Eigenmeasures and sharpness of the phase transition for the contact process

It is well known that for contact processes, extinction is exponentially fast in the whole subcritical regime. In this talk, I will present a new and surprisingly short proof of this fact that is based on eigenmeasures, which are possibly infinite measures on the set of nonempty configurations that are preserved under the dynamics up to a time-dependent exponential factor. In particular, in the subcritical regime, there is a one-to-one correspondence between translation-invariant eigenmeasures and quasi-invariant laws of the process modulo translations.

Balint Toth

Super-diffusivity of the periodic Lorentz-gas in the Boltzmann-Grad limit

We prove central limit theorem and invariance principle under superdiffusive scaling $\sqrt{t \log t}$ for the displacement of particle in the $Z^d$-based periodic Lorentz gas, in the Boltzmann-Grad limit. The result holds in any dimension.
(joint work with Jens Marklof (Bristol))

Stu Whittington

Self-avoiding walks and related polymer models

Self-avoiding walks are a standard model of the configurational properties of long linear polymer molecules in dilute solution in good solvents. They can be adapted to model polymer adsorption at a surface and polymer collapse in a poor solvent. The vertices of the walk can be coloured to model random copolymer phenomena such as localization at an interface between two immiscible solvents. Using micro-manipulation techniques polymers can be extended, or pulled from one phase to another, and self-avoiding walks are a useful model of such systems. This talk will give a general survey of what is known rigorously about these models, and will discuss several open questions.
 

PRACTICAL INFORMATION

●      Venue

Eurandom, 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).
Accessibility TU/e campus and map.

●      Registration

Registration is free, but compulsory for speakers and participants.

Speakers/Invitees please use this link to register: Registration speakers/invitees

Participants please use this link to register: Registration participants

●      Accommodation

For invited participants, we will take care of accommodation. Other attendees will have to make their own arrangements.

For 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.

●      Travel

For 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 or consult our map with highway connections.

●      Conference facilities : Conference room, Metaforum Building  MF11&12

The meeting-room 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 Secretariat

Upon 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 day-to-day running of the conference: registration, issuing certificates and receipts, etc.

●      Cancellation

Should you need to cancel your participation, please contact Patty Koorn, the Workshop Officer.

     ●      Contact

Mrs. Patty Koorn, Workshop Officer, Eurandom/TU Eindhoven, koorn@eurandom.tue.nl

SPONSORS

The organisers acknowledge the financial support/sponsorship of:

Last updated 21-12-16,
by PK

 P.O. Box 513, 5600 MB Eindhoven, The Netherlands
tel. +31 40 2478100  
  e-mail: info@eurandom.tue.nl