Monday  Tuesday  Wednesday  

09:00  9:50  L. Taggi 
V. Tassion 

9:50  10:20  Welcome (with coffee) 
Coffee break 
Coffee break 
10:20  11:10  N. Crawford 
A. Glazman 
M. Tassy 
11:20  12:10  R. Peled 
R. Peled 
M. Harel 
12:10  14:20  Lunch break 
Lunch break 
Lunch break 
14:20  15:10  E. Powell 
A. Elvey Price 
J. Aru 
15:10  15:40  Coffee break 
Coffee break 
Coffee break 
15:40  16:30  S. Goswami 
Open problems session 
M. Lis 
Juhan Aru
Extremal distance and conformal radius of a CLE_4 loop.
Conformal loop ensembles (CLE_\kappa) are random collections of loops in a simplyconnected planar domain. For 8/3 < \kappa \leq 4 they are characterised by conformal invariance + a certain Markovian property, and are conjectured to appear as scaling limits of different loop O(n) models at critical parameters. We consider CLE_4 in the unit disk and take a closer look at the loop L surrounding the origin. We will discuss how to calculate the law of the "distance" of L to the origin, the law of the "distance" of L to the boundary of the unit disk, and the joint law of both of these distances. This is joint work with Titus Lupu and Avelio Sepulveda.
Alexander Glazman
Sixvertex and AshkinTeller models: order/disorder phase transition.
The AshkinTeller model is a classical fourcomponent spin model introduced in 1943. It can be viewed as a pair of Ising models \(\tau\) and \(\tau'\) with parameter \(J\) that are coupled by assigning parameter \(U\) for the interaction of the products \(\tau*\tau'\) at every two neighbouring vertices. On the selfdual curve \(\sinh 2J = e^{2U}\), the AshkinTeller model can be coupled with the sixvertex model with parameters \(a=b=1,\, c=\coth 2J\) and is conjectured to be conformally invariant. The latter model has a heightfunction representation. We show that the height at a given face diverges logarithmically in the size of the domain when \(c=2\) and remains uniformly bounded when \(c>2\). In the latter case we obtain a complete description of translationinvariant Gibbs states and deduce that the AshkinTeller model on the selfdual line exhibits the following symmetrybreaking whenever \(J < U\): correlations of spins \(\tau\) and \(\tau'\) decay exponentially fast, while the product \(\tau*\tau'\) is ferromagnetically ordered.
The proof uses the BaxterKellandWu coupling between the sixvertex and the randomcluster models, as well as the recent results establishing the order of the phase transition in the latter model. However, in the talk, we will focus mostly on other parts of the proof:
 description of the heightfunction Gibbs states via heightfunction mappings and Tcircuits,
 coupling between the AshkinTeller and the sixvertex models via an FKIsingtype representation of these two models.
(this is joint work with Ron Peled)
Subhajit Goswami
Anomalous diffusion on the GFF landscape.
Diffusion in a random potential with logcorrelations is a model of great interest studied in different contexts in statistical mechanics, condensed matter physics and population biology among others. In twodimension a canonical example of a logcorrelated field is the Gaussian free field (GFF). In this talk I will present a family of random walk models on the square lattice indexed by an inverse temperature parameter where the the underlying transition probabilities are governed by a sample of the twodimensional GFF with appropriate boundary conditions. The random walk jumps to a neighbour v with probability proportional to the exponential of the product of inverse temperature and field value at v. As such this is a model of random walk in random environment (RWRE) where the underlying environment is strongly correlated. It has been predicted in the physics literature that this walk is subdiffusive and furthermore the diffusive exponent exhibits a dynamic phase transition around a certain critical temperature. I will discuss some rigorous results where we have been able to partially confirm these predictions including rigorous derivations of precise diffusive exponents at all temperatures.
Based on a joint work with Marek Biskup and Jian Ding.
Marcin Lis
Spins, percolation and height functions.
To capture apparent similarities in combinatorial representations of several twodimensional models of statistical mechanics,
we propose a fourparameter family of models (two integer and two real parameters)
which specializes to these known cases after a proper tuning.
The models come in the form of a spin system, a bond percolation model and a height function coupled together in a natural way
analogous to the EdwardsSokal coupling for the random cluster and Potts model.
We discuss basic properties of these models and their mutual relationships.
As special cases we recover the standard Potts and random cluster model, the sixvertex model and loop O(n) model,
the random current, double random current and XOR Ising model.
Moreover, based on a nonrigorous computation of Nienhuis of the variance of the height function in the sixvertex model,
we predict that the scaling limit of contours in the symmetric sixvertex model is a specific level set of the Gaussian free field.
This is a generalization of Wilson's conjecture about the scaling limit of the interfaces in the XOR Ising model.
Ellen Powell
A characterisation of the 2d Gaussian Free Field.
I will discuss a joint work with Nathanael Berestycki and Gourab Ray, in which we prove that a random distribution in two dimensions satisfying conformal invariance and a natural domain Markov property must be a multiple of the Gaussian free field. This result holds subject only to a mild moment assumption.
Ron Peled
The loop O(n) model  the main conjectures and some recent progress.
The loop O(n) model is a model for nonintersecting, selfavoiding loops. Its version on the hexagonal lattice is especially intriguing due to its conjectured critical behavior, on which many detailed predictions are available in the physics literature, and due to the fact that it generalizes several other models of interest. I will introduce the hexagonal lattice model, explain the main conjectures regarding it and the limited rigorous progress made. I will then focus on a recent work which uses the "XOR trick" to show the existence of macroscopic loops in a certain subset of the phase diagram. This is the first rigorous proof that macroscopic loops emerge for a positive area of the parameter space.
Joint work with Nicholas Crawford, Alexander Glazman and Matan Harel.
Ron Peled
Site percolation on planar graphs and circle packings.
Color each site of a planar graph blue with probability p and red with probability 1p, independently among sites. For which values of p is there an infinite connected component of blue sites? This is a classical problem in statistical physics and probability theory which is well studied when the planar graph is a lattice. Recently, Itai Benjamini made several conjectures on the problem for general planar graphs, relating it to the behavior of the simple random walk there. Understanding the problem for general planar graphs is also suggested by the study of the loop O(n) model. I will discuss a connection of the problem with circle packings and the progress that it yields.
Andrew Elvey Price
Counting planar maps equipped with a height function.
We define a height function on a graph to be a labelling of the vertices such that adjacent labels differ by exactly 1. We study planar quadrangulations equipped with a height function, which we call heightlabelled quadrangulations. EP and Guttmann showed that these are in bijection with 4valent maps equipped with an Eulerian orientation, which were counted exactly by Kostov in 1999 as a specialisation of what he called the six vertex model on a random lattice. I will describe an alternative method with which we count heightlabelled quadrangulations directly. This leads us to a much simpler form of the solution than what was found by Kostov. We also generalise our method to determine the exact distribution of labels amongst all heightlabelled quadrangulations.
This is joint work with Mireille BousquetMelou.
Lorenzo Taggi
Nondecay of correlations in the dimer model and phase transition in lattice permutations in Z^d, d>2, via reflection positivity.
In this talk we consider the dimer model and the model of random lattice permutations. Our first result is that the twopoint correlation function of the dimer model in Z^d does not decay with the distance when d > 2. This confirms physicists conjectures and shows that in dimensions greater than two the dimer model behaves drastically differently than in two dimensions, in which case it is known that correlations decay polynomially. Our second result involves a selfavoiding walk interacting via hardcore repulsion with a system of mutuallydisjoint (possibly oriented and coloured) selfavoiding loops and double dimers, a model which is related to the quantum Bose gas through a representation due to Feynman 1953 and to the loop O(N) model. We prove the occurrence of a phase transition in this model when d > 2, namely the existence of a regime in which the selfavoiding walk is 'long' and the distance between its endpoints grows linearly with the diameter of the box. These results follow from the derivation of an 'Infraredultraviolet' bound from a new probabilistic settings, with walks entering into the system from some 'virtual' vertices and interacting with coloured loops, and from the property of reflection positivity of such a gas of loops and walks.
Martin Tassy
Delocalization of uniform graph homomorphisms from \(Z^2\) to \(Z\).
Graph homomorphisms from the \(Z^2\) lattice to \(Z\) are functions on \(Z^2\) whose gradients equal \(1\) in absolute value. Following arguments presented by Scott Sheffield in Random surface which are adapted and simplified to the present settings, we will show that this model delocalizes in two dimensions, having no translationinvariant Gibbs measures for the uniform sampling subject to boundary conditions. We also obtain additional results in higher dimensions including the facts that every ergodic Gibbs measure is extremal. Finally we will discuss what part of the arguments can be generalized to other height function models such as the loop O(n) model.
Acommodation: Rooms have been reserved in Hotel du Faucon
Directions: Direct trains are available from Geneva or Zurich airports; it takes around 1h30 from either. Tickets and schedules are avaiblable online at www.cff.ch. The hotel is next to the station. The campus of the university is 15 minutes walking from the hotel. Alternatively, one may take a bus (numbers 1 or 3) to Perolles Charmettes; tickets for the bus are provided by the hotel. See this map.
Conference dinner:
All participants are invited to the conference dinner that will take place at the restaurant
Aux Tanneurs on Tuesday at 7pm.
Please use this form
to confirm your participation and choose the menu before Saturday 31st of August.
The original version of the menu may be found here (in French); the English translation is available on the form.
The restaurant is in the Place Petit Saint Jean, at the very bottom of the basse ville. See this
map.
Blackboard vs. beamer: Both blackboard and beamer talks are welcome.
Wireless internet: The hotel provides wifi. The university provides wifi through the eduroam system. Alternatively, ask the organiser to set up an account for you.