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# Franco-Dutch YEP Workshop

## May 28, 2018 - Jun 1, 2018

**"Mathematical Statistical Mechanics, Random Graphs and Related Topics"**

#### Sponsored by

*Aléa-Networks-Eurandom ***fellowships** for the Franco-Dutch YEP

To apply for support for travel/accommodation, please see: **GdRI CNRS AléaNetworks**

#### Summary

The model of the workshop will be the Young European Probalilists series (YEP) that are held annually in Eurandom, consisting of mini-courses and regular talks from both "junior" or "senior" researchers from the Dutch and French probabililist communities.

#### Organizers

The meeting is co-organized within the stochastic groups of TU Eindhoven and Université Paris-Est Créteil within their common CNRS affiliation, and intend to be the occasion to advert and celebrate the renewal of the CNRS-UMI affiliation of Eurandom.

Remco van der Hofstad (TU Eindhoven)

Arnaud Le Ny (CNRS UPEC Paris Est)

#### Speakers

The conference will contain two mini-courses of

- Pierre Picco (CNRS Marseille)

- Béatrice de Tilière (UPEC Paris-Est)

There will be talks from invited speakers, in and around the mini-course areas ; Currently confirmed speakers are

- Luca Avena (Leiden University)

- Rob van den Berg (CWI Amsterdam)

- Hao Can (Kyoto University)

- Aurélia Deshayes (Université Paris Diderot)

- Robert Fitzner (TU Eindhoven)

- Hakan Guldas (Leiden University)

- Alexandre Gaudillière (CNRS Marseille)

- Bénédicte Haas (Université Paris Nord)

- Frank den Hollander (Leiden University)

- Tim Hulshof (TU Eindhoven)

- Bruno Kimura (TU Delft)

- Irène Marcovici (Université de Lorraine)

- Cyril Marzouk (Université Paris-Sud)

- Frank Redig (TU Delft)

- Clara Stegehuis (TU Eindhoven)

- Daniel Valesin (University of Groningen)

#### Special Afternoon

On Friday afternoon, June 1, there will be a celebration of the extension of the CNRS label UMI that is awarded to Eurandom. We cordially invite you to attend this special afternoon.

#### Programme

#### Abstracts

**Luca Avena**

**Random Walks in Cooling Random Environment
**We propose a model of a one-dimensional random walk in dynamic random environment that interpolates betweentwo classical settings: (I) the random environment is sampled at time zero only and stays “frozen”; (II) the random environment is resampled at every unit of time. It is well known that in setting (II) strong homogenization takes place and the asymptotic behavior of the walk is as for an homogeneous Markov chain. While in setting (I) strong trapping phenomena are responsible for anomalous limiting behaviors (such as e.g. transient regime with zero speed, polynomial decay of rare events, non-diffusive fluctuations). In our interpolating “cooling” model, the random environment is resampled along an increasing sequence of deterministic times and we look at different resampling growing regimes. We characterize a.s. asymptotic speed, large deviations for the empirical speed and fluctuations under recurrence assumptions depending on how fast the resampling occurs. Proofs are based on a mixture of ergodic theory and concentration inequalities suited to the “cooling” model. Open problems will also be discussed.

(joint work with Y. Chino, C. da Costa and F. den Hollander)

**Rob van den Berg**

**Near-critical percolation with `heavy-tailed' impurities
**In the ordinary site percolation model with parameter p, the vertices of a lattice are, independently of each other, removed with probability 1-p.

Now suppose that in addition we also remove for each vertex (again independently of the others) with very small probability a (possibly very large) random box centered at the vertex. We focus on the case of the triangular lattice. It turns out that certain choices of the various parameters in this model give rise to delicate competing effects on the `global' connectivity properties of the remaining network. Although the study of these properties is interesting in itself, the main motivation comes from questions concerning quite natural processes, e.g. aggregation, spread of information and forest-fires.

(joint work with Pierre Nolin)

**Hao Can**

**Annealed Ising model on random regular graphs**

In a recent paper, Giardina, Giberti, Hofstad, Prioriello have proved a law of large numbers and a central limit theorem with respect to annealed measure of Ising model on some random graphs including random regular graph of degree 2. In this talk, we prove the limit theorems for random regular graphs with degree larger than 2. Moreover, we also study critical behavior of this model.

**Guillaume Conchon-Kerjean**

**Scaling limit of a configuration model with power-law degrees
**We prove that a critical configuration model with i.i.d. degrees, whose law has a power-tail behaviour with exponent α+1 for some α in (1,2), has a metric space scaling limit. To this purpose, we run through the graph in a depth-first manner, sampling an exploration process whose limit in law is absolutely continuous with respect to that of a spectrally positive α-stable L\'{e}vy process. So do its excursions above past minima, who then encode measure-changed α-stable trees, in which we make a random number of vertex-identifications to obtain the limit components of the graph.

(joint work with Christina Goldschmidt)

**Aurélia Deshayes**

**Front of the Friedrickson-Andersen one facilitated spin
**The Friedrickson-Andersen one facilitated spin (FA-1f) model is a kinetically contrained model (KCM) where each spin can flip provided at least one nearest neighbor is empty. KCM are non attractive interacting particle systems and the study of shapes can be complicated. We study the non equilibrium dynamics of FA-1f (in dimension 1) started from a configuration entirely occupied on the left half-line and focus on the evolution of the front, namely the position of the leftmost zero. We prove, for some flip parameters, a law of large numbers and a central limit theorem for the front, as well as the convergence to an invariant measure of the law of the process seen from the front.

(joint work with Oriane Blondel and Cristina Toninelli)

**Robert FItzner**

**Mean-field behavior using the lace expansion
**The lace expansion is a powerful perturbative technique to analyze the critical behavior of random spatial processes. Its major application is to prove that a model shows mean-field behavior in dimensions above the so-called upper critical dimension. We present recent result on the nearest-neighbor square lattice. Most prominently, F. and van der Hofstad proved that nearest-neighbor percolation shows mean field behavior in all dimension d>10.

For percolation, mean-field behavior can be loosely translated to the statement that their spatial correlation function behaves asymptotically like the Greens function of the simple random walk. This mean-field behavior can only be expected if the dimension of the underlying lattice is high enough. It is predicted that d>6 is high enough.

We explain idea of the lace expansion and explain the problem of applying the models in the dimension d=7,8,9,10.

(joint work with Remco van der Hofstad)

**Alexandre Gaudilliere**

**Asymptotic exponential law for the transition time to equilibrium of the metastable kinetic Ising model with vanishing magnetic field**

We make a pathwise description of the transition to equilibrium for the Glauber dynamic on a diverging size two-dimensional torus, at low enough temperature and with a magnetic field that scales as the square root of the inverse volume of the torus.

We use soft measures and compute soft capacities to do so.

**Hakan Güldas**

**Random walks on dynamic configuration models: a trichotomy
**In this talk, first I will introduce dynamic configuration model which is a dynamic random graph model in discrete time. Then, I will go into details of our results about mixing times of random walks on dynamic configuration model. The results I will give identify the behaviour of mixing times in terms of the proportion of edges that changes at every step of graph dynamics when the number of vertices is large.

(joint work with Luca Avena, Remco van der Hofstad and Frank den Hollander)

**Bénédicte Haas**

**Random trees constructed by aggregation
**We study a general procedure that builds random continuous trees by gluing recursively a new branch on a uniform point of the pre-existing tree. This encompasses the famous "line-breaking" construction of the Brownian tree of Aldous. Our aim is to see how the sequence of lengths of branches influences some geometric properties of the limiting tree, such as compactness, height and Hausdorff dimension. This is partly based on a joint work with Nicolas Curien (Université Paris-Sud). At the end of the talk I will also mention some extensions recently obtained by Delphin Sénizergues (Université Paris-Nord) of these results to the random gluing of (random) pointed metric spaces."

**Frank den Hollander**

**Spatially Extended Pinning
**We consider a directed polymer interacting with a linear interface. The

monomers carry random charges. Each monomer contributes an energy to the

interaction Hamiltonian that depends on its charge as well as its height

with respect to the interface, modulated by an interaction potential.

The configurations of the polymer are weighted according to the Gibbs

measure associated with the interaction Hamiltonian at a given inverse

temperature, where the reference measure is given by a recurrent Markov

chain.

We are interested in both the quenched and the annealed free energy per

monomer in the limit as the polymer becomes large. We find that each

exhibits a phase transition along a critical curve separating a localized

phase (where the polymer stays close to the interface) from a delocalized

phase (where the polymer wanders away from the interface). We obtain

variational formulas for the critical curves, and find that the quenched

phase transition is at least of second order. We obtain upper and lower

bounds on the quenched critical curve in terms of the annealed critical

curve. In addition, for the special case where the reference measure is

given by a Bessel random walk, we identify the weak disorder scaling limit

of the annealed free energy and the annealed critical curve in three

different regimes for the tail exponent of the interaction potential.

(joint work with Francesco Caravenna, University of Milano-Bicocca, Italy)

**Tim Hulshof**

**Higher order corrections for anisotropic bootstrap percolation
**Bootstrap percolation is a very simple model for growth from a random initial configuration on finite lattices. The model has many applications, for instance to model the spread of infections and magnets at low temperatures, to name two, but it is also interesting from a purely mathematical perspective. The model parameter has a critical value, at which the behaviour changes sharply. One interesting feature of bootstrap percolaton is a phenomenon called the “bootstrap paradox” which relates to a big discrepancy between numerical and theoretical estimates of the critical value of bootstrap percolation models.

I will discuss recent work in which we give the most accurate theoretical estimate for the critical value of any bootstrap model to date, and show how it (tentatively) resolves the paradox.

(joint work with Hugo Duminil-Copin and Aernout van Enter)

**Bruno Kimura**

**Phase transition in the one-dimensional long range Ising model with decaying external fields
**The long range Ising models differs from the classical ones by the fact that the spin-spin interaction decays spatially. Since long range models often behaves like short range models in higher dimension, we studied the phase diagram of one-dimensional Ising models in d=1 and proved that even in the presence of certain external fields there is phase transition at low temperatures. The techniques used to prove such result were based contour arguments developed by Froehlich, Spencer, Cassandro, Presutti et al.

**Irène Marcovici **

**Ergodicity of some classes of cellular automata subject to noise
**Cellular automata (CA) are dynamical systems on symbolic configurations on the lattice. They are also used as models of massively parallel computers. As dynamical systems, one would like to understand the effect of small random perturbations on the dynamics of CA. As models of computation, they can be used to study the reliability of computation against noise.

We consider various families of CA and prove that they are highly unstable against noise, meaning that they forget their initial conditions under slightest positive noise. This is manifested as the ergodicity of the resulting probabilistic CA. For instance, we prove that if the deterministic model is either nilpotent (highly stable) or permutive (highly chaotic), its small random perturbations are ergodic.

We are also interested in the sensitivity of tilings against noise, and I will briefly discuss the links between these two problems.

(joint work with Mathieu Sablik and Siamak Taati)

**Cyril Marzouk**

**Geometry of planar maps with large degrees
**In 2003, Angel & Schramm proved that a uniformly random triangulation of the sphere converges in distribution as the number of triangles tends to infinity towards an random infinite triangulation of the plane. This result has been then extended to so-called Boltzmann maps obtained loosely speaking by first sampling i.i.d. random polygons and then paving the sphere uniformly at random with these polygons. A particular case of interest which arises naturally is when the law on the sizes of the polygons has a heavy tail and precisely belongs to the domain of attraction of some stable law.

I will discuss the geometric properties of the dual graph of these stable infinite random maps (so the vertex-degrees have a heavy tail), with emphasis on the case of the 3/2-stable law for which such graphs exhibit an intermediate volume growth: the volume of the metric balls grows as the exponential of the square-root of the degree.

Based on a first work due to Timothy Budd (Radboud Universiteit, Nijmegen) & Nicolas Curien (Université Paris-Sud) and a second joint work with these two authors.

**Pierre Picco**

**Phase transitions in one dimension
**In this series of 3 talks, I will present a graphical representation of the spin configurations that allows to prove the existence of phase transition in one-dinensional Ising model with long range interactions by a Peierls argument.

Emphasis will be put on pictures and three extremal problems linked to this construction will be discussed.

**Frank Redig**

**Density fluctuations for the inclusion process in the condensation limit**

We consider the symmetric inclusion process, which is a system of particles performing random walks on the lattice $\Zd$ and interacting by attracting each other. In the limit of low random walk jump rate, this process shows condensation phenomena, i.e., large piles of particles are created. We look at how condensates emerge from a homogeneous initial product measure (coarsening).

We study this via the density correlation function and show how it scales to a quantity related to the local time of sticky Brownian motion.

The starting point of this analysis is self-duality and an exact formula for the transition probabilities of two inclusion particles.

(joint work with Gioia Carinci (Delft), Cristian Giardina (Modena))

**Clara Stegehuis**

**Optimal graphlet structures
**Subgraphs contain crucial information about network structure and function. For inhomogeneous random graphs with infinite-variance power-law degrees, we count the number of times a small connected graph occurs as an induced subgraph (graphlet counting). We introduce an optimization problem to identify the dominant structure of any given subgraph. The unique optimizer describes the degrees of the vertices that together span the most likely subgraph. We find that every subgraph occurs typically between vertices with specific degree ranges. In this way, we can count and characterize all subgraphs.

**Béatrice de Tilière**

**Aspects of the dimer and Z-invariant Ising models**

We plan to cover aspects of two models of statistical mechanics: the dimer model and the Z-invariant Ising model. Our goal for the dimer model is to describe the founding results of Kasteleyn, Temperley and Fisher, and explain the full description given by Kenyon, Okounkov and Sheffield in the infinite, bipartite, periodic case. Our next topic is the Z-invariant Ising model, introduced by Baxter. After having introduced the model, we will mention Fisher's correspondence allowing to study the planar Ising model through the dimer model on a decorated, non bipartite graph. We will then present results obtained in collaboration with Cédric Boutillier and Kilian Raschel proving local expressions for the free energy and probabilities of the Z-invariant Ising model, and establishing that the phase transition is the same as that of rooted spanning forests.

**Daniel Valesin**

**The asymmetric multitype contact process
**We study a class of interacting particle systems known as the multitype contact process on Z^d. In this model, sites of Z^d can be either empty or occupied by an individual of one of two species. Individuals die with rate one and send descendants to neighboring sites with a rate that depends on their (the parent's) type. Births are not allowed at sites that are already occupied. We assume that one of the types has a birth rate that is larger than that of the other type, and larger than the critical value of the standard contact process. We prove that, if initially present, the stronger type has a positive probability of never going extinct. Conditionally on this event, it takes over a ball of radius growing linearly in time. We also completely characterize the set of stationary distributions of the process and prove a complete convergence theorem.

(joint work with Pedro L. B. Pantoja and Thomas Mountford)

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#### Practical Information

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