This workshop is the ninth in a successful series of YEP meetings taking place at Eurandom in the past years. Traditionally, these meetings bring together junior probabilists from all over Europe working in a particular area of research, allowing them to present and discuss their recent work. This year, the main focus of the workshop will be on "Two dimensional statistical mechanics".

Since the introduction of SLE processes by Oded Schramm and the work of Stanislav Smirnov on conformal invariance of percolation, the area of two dimensional statistical mechanics has become one of the most exciting and active subfields of modern probability. Substantial progress has been made since then, especially in understanding two-dimensional phase transitions, and answers to many fundamental and long standing questions have been found. Apart from its direct relevance to physics, this new and exciting area of mathematics is underlied by deep connections between probability, combinatorics, the theory of conformal maps, and many other parts of mathematics, and continues to be a great source of mathematical discoveries.

The workshop will consist of three mini-courses and a number of talks by young European researchers working on different models of two-dimensional statistical mechanics, leaving plenty of time for discussions and interactions among the participants.

This year, the workshop is part of the second Stochastic Activity Month (SAM). During this month, Eurandom will host a number of workshops and scientific visitors working in the area of "Scaling limits in spatial probability". For more details: SAM2

ORGANISERS

FORMER YEP WORKSHOPS

SPEAKERS

Mini-courses

Extended talks

Invited speakers

PROGRAMME

MONDAY FEB 27

TUESDAY FEB 28

WEDNESDAY FEB 29

THURSDAY MAR 1

FRIDAY MAR 2

ABSTRACTS

MINI COURSES

Dmitry Chelkak

Discrete complex analysis on the microscopic level: conformal invariants without conformal invariance

Dealing with some 2D lattice model and its scaling limit (e.g., with the 2D Brownian motion in a fixed planar domain, which can be realized
as a limit of random walks on refining lattices $\delta\mathbb{Z}^2$), one usually works in the context when the lattice mesh $\delta$ tends
to zero. Then, one can argue that a pre-limiting behavior of the model is sufficiently close to the limiting one, if $\delta$ is small enough, e.g., the random walks hitting probabilities (discrete harmonic measures) become close to the Brownian motion hitting probabilities (classical harmonic measure) as $\delta\to 0$. After re-scaling by $\delta^{-1}$, such statements provide an information about random walk properties in (the bulk of) \emph{large} discrete domains in $\mathbb{Z}^2$. In this course, we are interested in uniform estimates which hold true
for \emph{arbitrary} discrete domains, possibly having \emph{many fiords and bottlenecks of various widths}, including very thin (several lattice steps) ones. Having in mind the classical geometric complex analysis as a guideline, we would like to construct its discrete version ``staying on a microscopic level'' (i.e., without any limit passage), which allows one to handle discrete domains by more-or-less the same methods as the classical (continuous) ones. The main objects of our interest are discrete quadrilaterals, i.e. simply connected domains $\Omega$ with four marked boundary points $a,b,c,d$. Focusing on quadrilaterals, we are motivated by two reasons. First, in the classical theory this is the ``minimal'' configuration which has a nontrivial conformal invariant (e.g., all simply connected domains with three marked boundary points are conformally equivalent due to the Riemann mapping theorem). Second, quadrilateral is an archetypical configuration for the 2D lattice models theory, where one often needs to estimate the probability of some crossing-type event between the opposite sides $[ab]$ and $[cd]$ of $\Omega$.
We prove a number of uniform double-sided estimates (``toolbox'') relating discrete counterparts of several classical conformal invariants of a configuration $(\Omega;a,b,c,d)$: cross-ratios, extremal lengths and random walk partition functions. This allows one to use classical methods of geometric complex analysis without any reference to geometric properties of $\Omega$. Applications include a discrete version of the classical ``$\int\frac{dx}{\theta(x)}$ estimate'' and some ``surgery technique'' developed for discrete quadrilaterals.

Nicolas Curien 

What is a random planar geometry? 

A planar map is proper embedding of a finite connected planar graph into the sphere. The theory of random planar maps has been developing over the last years in part motivated by the theory of two-dimensional quantum gravity. In particular, very recently, Le Gall and Miermont independently showed that a large class of random planar maps admits a continuum limit, a compact random surface called the Brownian map.
In this minicourse we will discuss in details uniform infinite planar maps which appear as local limits of uniform planar maps as their sizes go to infinity. As a key tool we will first discuss local limits of random trees.

PRESENTATION

Gabor Pete

Dynamical and near-critical percolation: many questions and many answers

Critical site percolation on the triangular lattice with mesh 1/n is a key example of having a conformally invariant scaling limit. Dynamical
percolation is the natural time evolution with critical percolation as stationary measure: every site of the lattice is switching between open and closed according to an independent exponential clock. It has been studied from three closely related points of view:
(1) How long does it take to change the macroscopic crossings that describe the scaling limit? In other words, how noise sensitive are
the crossing events?
(2) On an infinite lattice, are there random times with exceptional behavior, e.g., with an infinite cluster? In other words, which events are dynamically sensitive? How long do we have to wait for the first time when the cluster of the origin is infinite? How does the infinite cluster look like at this first exceptional time and how at a typical exceptional time?
(3) With a well-chosen rate r(n) for the exponential clocks, is there a scaling limit of the process, giving a Markov process on continuum configurations? In this scaling limit, what is the probability that the unit square has the left-right crossing all along [0,t], for large time t?

One can also consider an asymmetric dynamical process, where a site becomes open forever when its clock first rings. We can now ask: how
long does it take to leave the critical world and make the system very well-connected, i.e., how large is the near-critical window? And again, is there a scaling limit of this asymmetric process? Can we get an asymmetric (called "massive") version of SLE(6)?
I will survey the answers to most of the above questions, with several proofs in detail and a few open problems, from joint works with
Christophe Garban, Alan Hammond, Elchanan Mossel and Oded Schramm over the past few years.

At the end, I will mention a striking difference between percolation and critical FK random cluster models with q>1: while the interesting
time-scale for dynamical and near-critical percolation are the same, in Ising-FK, in the near-critical process changes happen much faster
than in the symmetric one, due to a fascinating self-organized mechanism with which new edges appear as the system moves out of criticality. This is joint work with Hugo Duminil-Copin and Christophe Garban.

PRESENTATION

MAIN TALKS

Dmitry Beliaev

Two-point Schramm's formula and SLE-8/3 bubbles 

Simmons and Cardy recently predicted a formula for the probability that the chordal SLE$_{8/3}$ path passes to the left of two points in the upper half-plane. We discuss a rigorous proof of   

their formula. Starting from this result, we derive explicit expressions for several natural connectivity functions for SLE$_{8/3}$ bubbles and, equivalently, Brownian bubbles, conditioned to be of   

macroscopic size. By passing to a limit with such a bubble we construct a certain (chordal) restriction measure and in this way obtain a proof of a formula for the probability that two given points   

are between two commuting SLE$_{8/3}$ paths. The one-point version of this result has been predicted by Gamsa and Cardy.

Loren Coquille

Gibbs measures of the 2d Ising and Potts models

Abstract : In the late 1970s, in two celebrated papers, Aizenman and Higuchi independently established that all infinite-volume Gibbs measures of the 2d Ising model are a convex combination of the two pure phases.
After introducing the relevant definitions and concepts needed to understand the physical content of this result, I will present a new approach to it, with a number of advantages:
(i) a finite-volume, quantitative analogue (implying the classical claim) is obtained;
(ii) the scheme of the proof seems more natural and provides a better picture of the underlying physical phenomenon;
(iii) this new approach is substantially more robust.
The extension to the Potts model is work in progress, I will finish by explaining what are the major technical and physical difficulties to overcome.
(joint works with Yvan Velenik, and Hugo Duminil-Copin, Dima Ioffe and Yvan Velenik) 

PRESENTATION

Alan Hammond

Boundary roughness in supercritical FK-percolation models

If supercritical percolation is conditioned so that the cluster containing the origin is finite but large, then we obtain a model of phase separation, in which a droplet - in this case, the cluster containing the origin - is suspended in an ambient environment - here, the unique infinite cluster in percolation. The droplet has a macroscopic profile approximating the Wulff shape, which is an analytic curve that minimizes a surface tension. We will discuss natural measures of fluctuation for the boundary of the droplet, and explain in outline how to derive exponents for the fluctuation.

Kostya Izyurov

Conformal invariance of spin correlations in the Ising model

We will descrbe the recent proof of conformal invariance of spin correlations in the critical planar Ising model. The main tools are discrete holomorphic spinor observables and results on convergence of soultions to corresponding discrete boundary value problems.
(joint work with Dmitry Chelkak and Clément Hongler)

Antti Kemppainen

Annulus crossing and regularity of random curves

Establishing a bound on a probability of an annulus crossing event is a natural way to show regularity of a random curve. In this talk I'll concentrate on my work with Stanislav Smirnov (Geneva and St.Petersburg). In brief, by this set of results, if a desired bound holds for a family probability measures on the space of curves, then this family is supported on curves which are Hölder regular as curves and as Loewner chains. This framework can be used for constructing scaling limits for random curves (showing the existence of subsequential limits) and especially as a part of a proof that some given sequence of random curves converges to a Schramm-Loewner evolution (SLE). Time-permitting I'll mention other applications to SLE related things.

Kalle Kytola

Interface in critical 2D Ising model with plus-minus-free boundary conditions

Consider the square lattice Ising model at its critical point in simply connected domains with a boundary. Split the boundary to a few pieces, and impose different boundary conditions on each. The questions addressed in this talk deal with the scaling limit in which a given domain is approximated by subgraphs of the square lattice with mesh size tending to zero. In the scaling limit conformal invariance properties are expected if the boundary conditions are combinations of plus, minus and free. A traditional interpretation of conformal invariance concerns correlation functions, and our first results are explicit expressions for some correlation functions of boundary spins in terms of the Riemann uniformizing map of the domain.
A different point of view to conformal invariance, initiated by Schramm, is to focus attention to random curves or interfaces. In the Ising model, the work of Smirnov and collaborators shows conformal invariance of two kinds of interfaces: an exploration path in the FK representation of Ising with plus-free boundary conditions tends to the chordal SLE(16/3) process, and a curve in the low temperature expansion with plus-minus boundary conditions tends to the chordal SLE(3) process. Our work uses the first of these results to obtain a generalization of the second. The generalization concerns an interface in the low temperature expansion with plus-minus-free boundary conditions. We will show how the expressions for correlation functions identify the limit of this curve as a variant of SLE(3) called the dipolar SLE(3). This generalization was first conjectured by Bauer & Bernard & Houdayer.
(joint work with Clément Hongler (Columbia University))

Zhongyang Li

1-2 Model, Dimers and Clusters

A 1-2 model is a probability measure on subgraphs of a hexagonal lattice, satisfying the condition that the degree of present edges at each vertex is either 1 or 2. We construct an explicit correspondence between the 1-2 model and the dimer model on a decorated graph, and derive a closed form for the probability of  paths of the 1-2 model on the infinite periodic hexagonal lattice. We prove that the behavior of infinite clusters is different for small and large local weights, which is an evidence of the existence of a phase transition.

Ioan Manolescu

Bond Percolation on Isoradial Graphs

The star-triangle transformation is used to obtain an equivalence extending over a set bond percolation models on isoradial graphs. Amongst the consequences are box-crossing (RSW) inequalities and the universality of alternating arms exponents (assuming they exist) for such models, under some conditions. In particular this implies criticality for these models.
(joint with Geoffrey Grimmett)

Anthony Metcalfe

Universality problems relating to lozenge tilings of a hexagon

In this talk we consider the set of lozenge tilings of a half-hexagon, with fixed tile positions in the final column, and an equivalent discrete
interlaced particle system in the plane. We impose the uniform distribution on the set of all such tilings, and show that the system is determinantal. We specialise to tilings of the regular hexagon. It is known that, as the size of the regular hexagon increases, a typical tiling has frozen regions of tiles near the corners, and a disordered region of approximately circular shape in the center. We consider the local asymptotic
behaviour of the tiles on the boundary of this disordered region as the size of the hexagon increases. We use steepest descent methods to show that the tiles behave asymptotically like a determinantal random point field with the Airy kernel.
(joint work with Kurt Johansson and Erik Duse)

François Simenhaus

Zero-temperature 2D Ising model and anisotropic curve-shortening flow

Let D be a simply connected, smooth enough domain of R^2. For L>0 consider the continuous time, zero-temperature heat bath dynamics for the nearest-neighbor Ising model on Z^2 with initial condition such that sigma_x=-1 if x\in LD and sigma_x=+1 otherwise. It is conjectured that, in the diffusive limit where space is rescaled by L, time by L^2 and L tends to infinity, the boundary of the droplet of ``-'' spins follows a deterministic anisotropic curve-shortening flow, where the normal velocity at a point of its boundary is given by the local curvature times an explicit function of the local slope. We prove this conjecture at zero temperature when D is convex.
A preprint is available at http://arxiv.org/abs/1112.3160
(joint work with Hubert Lacoin and Fabio Lucio Toninelli)

Vincent Tassion

The critical value function in the divide and color model

The divide and color model is a simple and natural stochastic model for dependent colorings of the vertex set of an infinite graph. This model has two parameters: an edge-parameter p, which determines how strongly the states of different vertices depend on each other, and a coloring parameter r, which is the probability of coloring a given vertex red. For each value of p, there exists a critical coloring value R such that there is almost surely no infinite red cluster for all r infinite red cluster exists with positive probability for all r>R. In this talk, I will discuss some new results, obtained jointly with András Bálint and Vincent Beffara, concerning different properties, such as (non-)continuity and (non-)monotonicity, of the critical coloring value as a function of the edge-parameter.

Ariel Yadin

A Brief Introduction to SLE

This talk is aimed to be a brief introduction to Schramm-Loewner Evolution, or SLE.  The goal is to introduce the ideas leading to the definition, rather than prove the specific results.
We will start historically by introducing the Loewner equation, and then jumping forward 70 years to the domain Markov property and conformal invariance.  We will view these notions through the first process considered by Oded Schramm when he introduced SLE:  loop-erased random walk.
Time permitting we will extend some of these ideas to a discussion regarding the Laplacian-b random walk (which will be introduced during the talk).

SHORT TALK SESSIONS

Simon Aumann

Singularity of Nearcritical Percolation Scaling Limits

Nolin and Werner showed that a nearcritical percolation scaling limit is singular to SLE(6). In this talk we explain how their result can be generalised. We show that even two different nearcritical percolation scaling limits are singular with respect to each other.

Igor Kortchemski

The Brownian triangulation: a universal limit for random non-crossing configurations

We are interested in various models of random non-crossing configurations consisting of diagonals of convex polygons, such as uniform triangulations, dissections, non-crossing partitions or non-crossing trees. For all these models, we prove convergence in distribution towards Aldous’ Brownian triangulation of the disk. This has interesting combinatorial applications. 
(joint work with Nicolas Curien)

Marcin Lis

Computation of the critical temperature of the 2D Ising model via the combinatorial method

The combinatorial method expresses the partition sum of the Ising model as an exponential of a certain generating function of signed loops on the graph. We will show how to use this formula to obtain the critical temperature of the model as far as the analyticity of the thermodynamic limit of the free energy density is concerned. We will also briefly discuss the way to localize the critical point in terms of the behaviour of the correlation function.

Tobias Muller

Continuum bootstrap percolation

We consider a Poisson process of intensity lambda on the ball B(0,R) of radius R around the origin in d-dimensional space. Initially, we place a ball of radius one around each point of the Poisson process. Next, we apply the following rule until exhaustion: If two balls intersect then we remove both of them and replace them with the smallest ball that contains both.
We are interested in question whether the entire B(0,R) will eventually get covered or the process will stop before this happens. We show a "sharp threshold" for the probability of complete coverage when the intensity lambda of the Poisson process is const /  (ln R)^{d-1}.
(joint work with Anne Fey)

Francesca Nardi

Metastability for Kawasaki dynamics at low temperature with two types of particles

We study a two-dimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics at low temperature in a large finite box with an open boundary. Each pair of particles occupying neighboring sites has a negative binding energy provided their types are different, while each particle has a positive activation energy that depends on its type. There is no binding energy between neighboring particles of the same type. At the boundary of the box particles are created and annihilated in a way that represents the presence of an infinite gas reservoir. We start the dynamics from the empty box and compute the transition time to the full box. This transition is triggered by a critical droplet appearing somewhere in the box.

Ryokichi Tanaka

Large deviation on a covering graph with group of polynomial growth

We discuss a large deviation principle of a periodic random walk on a covering graph with its transformation group of polynomial volume growth. We shall observe, the behavior of a random walk at infinity is closely related to the Gromov-Hausdorff limit of an infinite graph and in our cases, the Carnot-Carath¥'{e}odory metric shows up in its limit space.

Christoph Temmel

Shearer's measure and stochastic domination of Bernoulli product fields

Let G=(V,E) be a locally finite graph. Let \vec{p}\in[0,1]^V. We show that Shearer's measure, introduced in the context of the Lovasz Local
Lemma, with marginal distribution determined by \vec{p} exists on G iff every Bernoulli random field with the same marginals and dependency graph G dominates stochastically a non-trivial Bernoulli product field. Additionally we derive a non-trivial uniform lower bound for the parameter vector of the dominated Bernoulli product field. This generalizes previous results by Liggett, Schonmann & Stacey in the
homogeneous case, in particular on the k-fuzz of Z. Using the connection between Shearer's measure and a hardcore gas established by Scott & Sokal, we apply bounds derived from cluster expansions of lattice gas partition functions to the stochastic domination problem.

PARTICIPANTS

PRACTICAL INFORMATION

Conference Location
The workshop location is Eurandom,  Den Dolech 2, 5612 AZ Eindhoven, Laplace Building, 1st floor, LG 1.105.

Eurandom is located on the campus of Eindhoven University of Technology, in the 'Laplacegebouw' building' (LG on the map). The university is located at 10 minutes walking distance from Eindhoven railway station (take the exit north side and walk towards the tall building on the right with the sign TU/e).

For all information on how to come to Eindhoven, please check http://www.eurandom.tue.nl/contact.htm

CONTACT
For more information please contact Mrs. Patty Koorn,
Workshop officer of  EURANDOM

OTHER SPONSORS

Research Network Programme: RGLIS - Random Geometry of Large Interacting Systems and Statistical Physics

Last updated 01-07-13,
by PK