*Moderator: *Hugo Duminil-Copin (Genève)

Michael Aizenman (Princeton),
Dmitry Chelkak (Genève & St.Petersburg),
Hugo Duminil-Copin (Genève),
Christophe Garban (Lyon),
Alessandro Giuliani (Roma),
Ioan Manolescu (Fribourg),
Vincent Tassion (Genève).

In this talk, I will discuss the connection between spin-models and percolation-type models. I will also introduce the notion of phase transition in these models and provide the audience with the general context for the results presented in other talks of the day.

**Michael Aizenman** (Princeton):
*Topological roots of the fermionic structures seen in the 2D Ising
model.*

**Dmitry Chelkak** (Genève & St.Petersburg):
*Magnetization in the layered Ising model.*

In recent years, a number of rigorous convergence results for the critical 2D Ising model correlation functions has been established via a careful analysis of various boundary value problems for fermionic observables, which satisfy a version of the discrete Cauchy-Riemann relations at the critical temperature. Since similar linear relations hold true for arbitrary interaction constants, one can also use them to derive some information on the 2D Ising model in more general setups. Following this route, we present a new formula for the magnetization (average value of a particular spin) in the `layered' Ising model considered in the discrete half-plane (above, 'layered' means that interaction constants depend on the distance to the boundary only). The answer is given in terms of truncated determinants of the square root of a simple Jacobi matrix constructed from a sequence of interaction constants, and leads to some natural conjectures on the decay of the magnetization at infinity depending on the choice of a sequence of coupling constants. Interestingly, this formula also gives an explicit answer at the criticality, which seems to have been unknown before. This is a joint work with Clement Hongler (EPFL, Lausanne).

**Christophe Garban** (Lyon):
*Exceptional times for critical percolation under conservative dynamics.*

Start with an initial critical percolation configuration w_0 in the plane. Let this configuration evolve in time according to a simple exclusion process with kernel P(x,y)\sim |x-y|^{-2-\alpha}. In a joint work with Hugo Vanneuville, we prove that if the long-range exponent \alpha is chosen sufficiently small, then there exists *exceptional times* t for which an infinite cluster appears in w_t (the result holds for conservative site dynamics on the triangular lattice). The existence of such exceptional times for i.i.d dynamics (where sites evolve according to independent Poisson Point process) goes back to the influential paper by Schramm-Steif in 2006. Here, to handle such a conservative case, we push further the analysis of *exclusion noise sensitivity* which had been initiated in Broman-Garban-Steif.

**Alessandro Giuliani** (Roma):
*Periodic striped ground states in Ising models with competing interactions.* pdf

We consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let J be the ratio between the strength of the ferromagnetic to antiferromagnetic interactions. The competition between these two kinds of interactions induces the system to form domains of minus spins in a background of plus spins, or vice versa. If the decay exponent p of the long range interaction is larger than d+1, with d the space dimension, this happens for all values of J smaller than a critical value J_c(p), beyond which the ground state is homogeneous. In this talk, we give a characterization of the infinite volume ground states of the system, for p>2d and J in a left neighborhood of J_c(p). In particular, we report a proof that the quasi-one-dimensional states consisting of infinite stripes (d=2) or slabs (d=3), all of the same optimal width and orientation, and alternating magnetization, are infinite volume ground states. We shall explain the key aspects of the proof, which is based on localization bounds combined with reflection positivity. Joint work with Robert Seiringer.

**Ioan Manolescu** (Fribourg):
*Bond Percolation on Isoradial Graphs.*
pdf

The star-triangle transformation is used to obtain an equivalence extending over a set of 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 work with Geoffrey Grimmett). If time permits, we will discuss ongoing work to generalise the result to certain random-cluster models on isoradial graphs (the latter is joint work with H. Duminil-Copin and J.H. Li).

**Vincent Tassion** (Genève):
*Critical behavior of Fortuin-Kasteleyn percolation in two dimensions.*

We will describe a theory of renormalization for crossing probabilities in planar dependent percolation models. We will discuss in details the case of the Fortuin-Kasteleyn percolation on the square lattice.