Talks (titles and abstracts)
Juhan Aru: tba
Barbara Dembin: tba
Philip Easo: tba
Christophe Garban: Invisibility of the integers for the discrete Gaussian chain via a Caffarelli-Silvestre extension of the discrete fractional Laplacian
The "Discrete Gaussian Chain" is a model which extends the celebrated long-range 1D lsing model with \(1/r^\alpha\) interactions. The latter model is known to have a rather intriguing phase-diagram. Instead of having +/- spins, the discrete Gaussian Chain is a random field with values in the integers Z.
After introducing this model and its history, I will describe its large scale fluctuations (described by fractional Brownian motion) and will compare its phase diagram with the case of long-range Ising model.
Alexander Glazman: FKG in height and colouring models
I will speak about the FKG property and its applications to different models: delocalisation of Lipschitz functions and graph homomorphisms through an FKG property (j.w. Lammers); Lipschitz functions on rhombic tilings with the Yang-Baxter weights (j.w. Beck-Tiefenbach, Dober); Baxter-Wu model and proper four-colourings of different lattices (j.w. Rax, Willmann). Time permitting, there will be also a riddle about graph homomorphisms to the triangular lattice - it can also be postponed to the open problem session.
Titus Lupu: 2D Brownian loop soups, conformally invariant fields and height gap
In my talk I will present the joint work with Antoine Jego and Wei Qian, also related to earlier works with Aru and Sepulveda, as well as with Aidekon, Berestycki, Jego. The Brownian loop soups are natural Poisson collections of Brownian loops in a 2D domain, that satisfy a conformal invariance property. In particular, there is a scale invariance property with tiny loops at all small scales. The question of clusters formed by these Brownian loops hase been first studied by Sheffield and Werner. They showed a phase transition at the intensity parameter c=1. For c<= 1, the outer boundaries have been identified as Conformal Loop Ensembles CLE_k. Later, in my joint work with Aru and Sepulveda, the whole clusters at c=1 have been identified as sign components of the 2D Gaussian free field. In the work with Jego and Qian, we focus on the subcritical regime c<1. We show that the one-arm probabilities for clusters behave like | log epsilon|^{-(1-c/2)}. We further show that out of the Minkowski contents of these clusters plus some independent signs, one can construct conformally invariant random fields, which are correlated as a power of log. We prove that all these fields present a height gap phenomenon, similarly to the 2D GFF. We also formulate conjectures on the renormalized powers of these fields.
Ioan Manolescu: Scaling relations for 2D FK-percolation
In 1987, Kesten related the critical and near-critical regimes of planar Bernoulli percolation proving the so-called scaling relations. He proved that the values of the one- and four-arm exponents determine (almost) all other critical exponents, such as that of the correlation length or of the percolation probability. His work relies crucially on the interpretation of derivatives of increasing events using pivotal edges.
We extend Kesten's work to FK-percolation (a.k.a. the random-cluster model) on the square lattice Z^2. This model has been shown to have a much more varied phase transition than Bernoulli percolation: continuous when the cluster parameter q lies between 1 and 4 and discontinuous for q above 4. The model is increasing in its edge-intensity, but increasing couplings are much more complex than for Bernoulli percolation and derivatives of increasing events are harder to control.
We introduce a concept replacing pivotality (with properties such as quasi-multiplicativity and stability in the critical window), which governs the derivative in FK-percolation, but which is shown to have a different critical exponent than the four-arm event. This new critical exponent replaces the four-arm exponent in Kesten's scaling relations for 1 < q \leq 4 and is determined by the critical scaling limit.
Based on joint work with Hugo Duminil-Copin
Stephen Muirhead: On the critical phase of strongly correlated percolation models
We consider nearest neighbour Euclidean percolation models for which correlations decay as a power law with exponent \(\alpha > 0\). Except for the integrable model of the metric graph GFF in \(d \ge 3\), the critical phase of these models is not well understood, even by physicists. It has been conjectured that the correlation length exponent is \(\nu = 2/\alpha\) if \(\alpha \le 2/\nu_{Bernoulli}\), and that critical two-dimensional models have a conformally invariant scaling limit. For a general class of models in two-dimensions we show that (i) scaling limits, if they exist, are non-degenerate in the sense that they contain macroscopic interfaces, and (ii) \(\nu = 2/\alpha + O(1)\) as \(\alpha \to 0\), and moreover \(\nu = 2/\alpha\) for \(\alpha < \alpha_0\) assuming the scaling relation \(\beta = \nu \rho_1\) holds.
Romain Panis: What insights does the mixing of random currents provide about the critical Ising model?
Aizenman and Duminil-Copin recently solved a long-standing open problem on the marginal triviality of the scaling limits of the Ising model in dimension four. A pivotal step of their argument is the derivation of a mixing property for the model's random current representation.
In this talk, I will discuss other consequences of this mixing property, which include the construction of the Incipient Infinite Cluster (IIC) of the FK-Ising model in dimensions d>2, the derivation of exact asymptotics for the susceptibility, and the proof of convergence to the Gaussian Free Field in dimensions d>4.
Pierre-François Rodriguez: Anatomy of clusters in percolation models with long-range dependence
The talk will discuss recent progress concerning the phase transition associated to percolation models with long-range dependence. These models originate on the one hand in works of Lebowitz-Saleur and Lebowitz-Bricmont-Maes from the mid 80’s, and in various fragmentation problems for random walks on the other. The latter, prompted in more recent works of Dembo-Sznitman and Benjamini-Sznitman (mid 00’s), touch on rather fundamental questions concerning the nature of random walk traces and their complements in higher dimensions.
Franco Severo: On the supercritical phase of the Phi4 model
The Phi4 model is a real-valued spin system with quartic potential. This model has deep connections with the classical Ising model, and both are expected to belong to the same universality class. In particular, the sign of the Phi4 model is an Ising model in a random environment determined by its absolute value, which naturally leads to a random cluster representation of the model. For this representation, we prove that local uniqueness of macroscopic cluster holds throughout the supercritical phase, which serves as the crucial step towards a detailed description of the supercritical behaviour through renormalisation techniques. The proof builds on certain tools shared by the Phi4 and Ising models, such as the Ginibre inequality and a random current representation. However, the unboundedness of spins in the Phi4 model imposes considerable difficulties when compared with the corresponding result for the Ising model.
Based on an ongoing work Trishen Gunaratnam, Christoforos Panagiotis and Romain Panis.
Augusto Teixeira: tba
Diederik Van Engelenburg: Pfaffians in spins models and percolation
It is well known that the Ising model on a planar graph satisfies so-called Pfaffian relations, but are there any other (natural) models that also have this property? I will present an answer to this problem: no, up to local (and rather explicit) modifications of the graph, "any" model with a Pfaffian correlation structure has to be both planar and Ising.
The main tool is the development of two interacting (or not) FK representations, if time permits I may also say a few words about other applications. Based on joint work with Marcin Lis.
Hugo Vanneuville: Volume or diameter?
Consider Bernoulli percolation. Is it easier to study the volume or the diameter of a cluster? Of course, it depends on the context, but I will try to motivate this naive question and illustrate it with examples, in particular the sharpness of the phase transition and the uniqueness of the infinite cluster.