Title: On the monotonicity of the speed of biaised random walk on a Galton-Watson tree without leaves.

Abstract: We will present different results related to the speed of biased random walks in random environments. Our focus will be on a recent paper by Ben Arous, Fribergh and Sidoravicius proving that the speed of the biased random walk on a Galton-Watson tree without leaves is increasing for high biases. This partially solves a question asked by Lyons, Pemantle and Peres.

Title: Search Games, The Cauchy process and Optimal Kakeya Sets

Abstract: A planar set that contains a unit segment in every direction is called a Kakeya set. These sets have been studied intensively in geometric measure theory and harmonic analysis since the work of Besicovich (1928); we find a new connection to game theory and probability via a search game first analyzed by Adler et al (2003). A hunter and a rabbit move on the n-vertex cycle without seeing each other. At each step, the hunter moves to a neighboring vertex or stays in place, while the rabbit is free to jump to any node. Thus they are engaged in a zero sum game, where the payoff is the capture time. The known optimal randomized strategies for hunter and rabbit achieve expected capture time of order n log n. We show that every rabbit strategy yields a Kakeya set; the optimal rabbit strategy is based on a discretized Cauchy random walk, and it yields a Kakeya set K consisting of 4n triangles, that has minimal area among such sets (the area of K is of order 1/log(n)). Passing to the scaling limit yields a simple construction of a random Kakeya set with zero area from two Brownian motions. (Joint work with Y. Babichenko, R. Peretz, P. Sousi and P. Winkler).

Tutorial Seminar: What is the mixing time for random walk on a graph?

Abstract: Consider a simple random walk on a finite graph. The mixing time is the time it takes the walk to reach a position that is approximately independent of the starting point; it has been studied intensively by combinatorialists, computer scientists and probabilists; the mixing time arises in statistical physics as well. Applications of mixing times range from random sampling and card shuffling, to understanding convergence to equilibrium in the Ising model. It is closely related to expansion and eigenvalues. Besides introducing this topic, I will also describe the open problem of understanding which random walks exhibit "cutoff", a sharp transition to stationarity first discovered by Diaconis, Shashahani and Aldous in the early 1980s but still mysterious.

Title: Convergence to equilibrium for nonreversible diffusions.

Abstract: The problem of convergence to equilibrium for diffusion processes is of theoretical as well as applied interest, for example in nonequilibrium statistical mechanics and in statistics, in particular in the study of Markov Chain Monte Carlo (MCMC) algorithms. Powerful techniques from analysis and PDEs, such as spectral theory and functional inequalities (e.g. logarithmic Sobolev inequalities) can be used in order to study convergence to equilibrium. Quite often, the diffusion processes that appear in applications are degenerate (in the sense that noise acts directly to only some of the degrees of freedom of the system) and/or nonreversible. The study of convergence to equilibrium for such systems requires the study of non-selfadjoint, possibly non-uniformly elliptic, second order differential operators. In this talk we show how the recently developed theory of hypocoercivity can be used to prove exponentially fast convergence to equilibrium for such diffusion processes. Furthermore, we will show how the addition of a nonreversible perturbation to a reversible diffusion can speed up convergence to equilibrium. This is joint work with M. Ottobre, K. Pravda-Starov, T. Lelievre and F. Nier.

This is a special event. Billingsley Lectures on Probability in honor of Professor Billingsley.

Title: Random Walk with Reinforcement

Abstract: Picture a triangle, with vertices labeled A, B, C. A random walker starts at A and chooses a random nearest neighbor. At each stage, the walker adds 1 to the weight of each crossed edge and chooses the next step with probability proportional to the current edge weights. The question is 'what happens?'. This simple problem leads into interesting corners: to Bayesian analysis of the transition mechanism of Markov chains (and protein folding) and to the hyperbolic sigma model of statistical physics. Work of (and with) Billingsley, Baccalado, Freedman, Tarres, and Sabot will be reviewed.

Title: Exchangeable random measures

Abstract: Classical theorems of de Finetti, Aldous-Hoover and Kallenberg describe the structure of exchangeable probability measures on spaces of sequences or arrays. Similarly, one can add an extra layer of randomness, and ask after exchangeable random measures on these spaces. It turns out that those classical theorems, coupled with an abstract version of the `replica trick' from statistical physics, give a structure theorem for these random measures also. This leads to a new proof of the Dovbysh-Sudakov Theorem describing exchangeable positive semi-definite matrices.

Title: Maximum independent sets in random d-regular graphs

Abstract: Satisfaction and optimization problems subject to random constraints are a well-studied area in the theory of computation. These problems also arise naturally in combinatorics, in the study of sparse random graphs. While the values of limiting thresholds have been conjectured for many such models, few have been rigorously established. In this context we study the size of maximum independent sets in random d-regular graphs. We show that for d exceeding a constant d(0), there exist explicit constants A, C depending on d such that the maximum size has constant fluctuations around A*n-C*(log n) establishing the one-step replica symmetry breaking heuristics developed by statistical physicists. As an application of our method we also prove an explicit satisfiability threshold in random regular k-NAE-SAT. This is joint work with Jian Ding and Allan Sly.

Title: Scaling limit of the abelian sandpile

Abstract: Which functions of two real variables can be expressed as limits of superharmonic functions from (1/n)Z² to (1/n²)Z? I'll discuss joint work with Wesley Pegden and Charles Smart on the case of quadratic functions, where this question has a surprising and beautiful answer: the maximal such quadratics are classified by the circles in a certain Apollonian circle packing of the plane. I'll also explain where the question came from (the title is a hint!).

Title Law of the extremes for the two-dimensional discrete Gaussian Free Field

Abstract: A two-dimensional discrete Gaussian Free Field (DGFF) is a centered Gaussian process over a finite subset (say, a square) of the square lattice with covariance given by the Green function of the simple random walk killed upon exit from this set. Recently, much effort has gone to the study of the concentration properties and tail estimates for the maximum of DGFF. In my talk I will address the limiting extreme-order statistics of DGFF as the square-size tends to infinity. In particular, I will show that for any sequence of squares along which the centered maximum converges in law, the (centered) extreme process converges in law to a randomly-shifted Gumbel Poisson point process which is decorated, independently around each point, by a random collection of auxiliary points. If there is any time left, I will review what we know and/or believe about the law of the random shift. This talk is based on joint work with Oren Louidor (UCLA).

Title: The Virasoro algebra and discrete Gaussian free field

Abstract: The Virasoro algebra is an infinite dimensional Lie algebra that plays an important role in the Conformal Field Theory (CFT) methods employed by physicists to describe and study conformally invariant scaling limits of planar critical lattice models from statistical physics. Despite much progress in the last decade, it seems fair to say that from a mathematical perspective many aspects of the connections between discrete model and continuum limit CFT remain somewhat mysterious. In the talk I will discuss recent joint work with C. Hongler and K. Kytola concerning the discrete Gaussian free field on a square grid. I will explain how for this model discrete complex analysis can be used to construct explicit (exact) representations of the Virasoro algebra of central charge 1 directly on the discrete level.

Title: Markov type and the multi-scale geometry of metric spaces

Abstract: The behavior of random walks on metric spaces can sometimes be understood by embedding such a walk into a nicer space (e.g. a Hilbert space) where the geometry is more readily approachable. This beautiful theme has seen a number of geometric and probabilistic applications. We offer a new twist on this study by showing that one can employ mappings that are significantly weaker than bi-Lipschitz. This is used to answer questions of Naor, Peres, Schramm, and Sheffield (2004) by proving that planar graph metrics and doubling metrics have Markov type 2. The main new technical idea is that martingales are significantly worse at aiming than one might at first expect. Joint work with Jian Ding and Yuval Peres.

Title: The Renormalization Group for Disordered Systems

Abstract: We investigate the Renormalization Group (RG) approach in finite- dimensional glassy systems, whose critical features are still not well-established, or simply unknown. We focus on spin and structural-glass models built on hierarchical lattices, which are the simplest non-mean-field systems where the RG framework emerges in a natural way. The resulting critical properties shed light on the critical behavior of spin and structural glasses beyond mean field, and suggest future directions for understanding the criticality of more realistic glassy systems.

Title: Geodesics and Direction in 2d First-Passage Percolation

Abstract: I will discuss geodesics in first-passage percolation, a model for fluid flow in a random medium. There are numerous conjectures about the existence, coalescence, and asymptotic direction of infinite geodesics under the model's random metric. C. Newman and collaborators have proved some of these under strong assumptions. I will explain recent results with Michael Damron which develop a framework for addressing these questions; this framework allows us to prove versions of Newman's results under minimal assumptions.

Title: Gaussian Free Field fluctuations for general-beta random matrix ensembles.

Abstract: It is now known that the asymptotic fluctuations of the height function of uniformly random lozenge tilings of planar domains (equivalently, stepped surfaces in 3d space) are governed by the Gaussian Free Field (GFF), which is a 2d analogue of the Brownian motion. On the other hand, in certain limit regimes such tilings converge to various random matrix ensembles corresponding to beta=2. This makes one wonder whether GFF should also somehow arise in general-beta random matrix ensembles. I will explain that this is indeed true and the asymptotics of fluctuations of classical general-beta random matrix ensembles is governed by GFF. This is joint work with A.Borodin.

Title: About heavy tailed random matrices.

Abstract:We investigate the behaviour of matrices which do not belong to the universality class of Wigner matrices because their entries have heavy tails.

Title: Chaos problem in mean field spin glasses

Abstract: The main objective in spin glasses from the physical perspective is to understand the strange magnetic properties of certain alloys. Yet the models invented to explain the observed phenomena are also of a rather fundamental nature in mathematics. In this talk we will first introduce the famous Sherrington-Kirkpatrick model as well as some known results about this model such as the Parisi formula and the limiting behavior of the Gibbs measure. Next, we will discuss the problems of chaos in the mixed p-spin models and present mathematically rigorous results including disorder, external field, and temperature chaos.

Abstract: This seminar was canceled. It will be rescheduled.

Title: The 2D SOS Model

Abstract: We present new results on the (2+1)-dimensional Solid-On-Solid model at low temperatures. Bricmont, El-Mellouki and Froelich (1986) showed that in the presence of a floor there is an entropic repulsion phenomenon, lifting the surface to a height which is logarithmic in the side of the box. We refine this and establish the typical height of the SOS surface is precisely the floor of [1/(4\beta)\log n], where n is the side-length of the box and \beta is the inverse-temperature. We determine the asymptotic shape of the top plateau and show that its boundary fluctuation are n^{1/3+o(1)}. Based on joint works with Pietro Caputo, Eyal Lubetzky, Fabio Martinelli and Fabio Toninelli.

Title: Spiking the random matrix hard edge.

Abstract: The largest eigenvalue of a finite rank perturbation of a random hermitian matrix is known to exhibit a phase transition (in the infinite dimensional limit). If the perturbation is small one sees the famous Tracy-Widom law, while a large perturbation results in a Gaussian fluctuation. In between there exists is a scaling window about a critical perturbation value leading to a separate family of limit laws. This basic discovery is due to Baik, Ben Arous, and Peche. More recently Bloemendal and Virag have shown this picture persists in the context of the general beta ensembles, giving new formulations of the critical limit laws . Yet another route, explained here, is to go through the random matrix hard edge, perturbing the smallest eigenvalues in the sample covariance set-up. A limiting procedure then recovers all the alluded to distributions. (Joint work with Jose Ramirez.)

Title: Directed polymers and the stochastic heat equation

Abstract: We show how some properties of the solutions of the Stochastic Heat Equation (SHE) can be derived from directed polymers in random environment. In particular, we show: * A new proof of the positivity of the solutions of the SHE * Improved bounds on the negative moments of the SHE * Results on the fluctuations of the log of the SHE in equilibrium, namely, the Cole-Hopf solution of the KPZ equation (if time allows).

Title: Second-order Boltzmann-Gibbs principle and applications

Abstract: The celebrated Botzmann-Gibbs principle introduced by Rost in the 80's roughly says the following. For stochastic systems with one or more conservation laws, fluctuations of the non-conserved quantities are faster than fluctuations of the conserved quantities. Therefore, in the right space-time window, the space-time fluctuations of a given observable are asymptotically equivalent to a linear functional of the conserved quantities. In one dimension, we prove two generalizations of this principle: a non-linear (or second-order) and a local version of it. This result opens a way to show convergence of fluctuations for non-linear models, like the ones on the fashionable KPZ universality class. As a corollary, we prove new convergence results for various observables of the asymmetric exclusion process, given in terms of solutions of the KPZ equation. Joint work with Patricia Gonçalves.

Title: SLE curves and natural parametrization

Title: Robust Gaussian noise stability

Abstract: Given two Gaussian vectors that are positively correlated, what is the probability that they both land in some fixed set A? Borell proved that this probability is maximized (over sets A with a given volume) when A is a half-space. We will give a new and simple proof of this fact, which also gives some stronger results. In particular, we can show that half-spaces uniquely maximize the probability above, and that sets which almost maximize this probability must be close to half-spaces.

Title: Spiking the random matrix hard edge.

Abstract: The largest eigenvalue of a finite rank perturbation of a random hermitian matrix is known to exhibit a phase transition (in the infinite dimensional limit). If the perturbation is small one sees the famous Tracy-Widom law, while a large perturbation results in a Gaussian fluctuation. In between there exists is a scaling window about a critical perturbation value leading to a separate family of limit laws. This basic discovery is due to Baik, Ben Arous, and Peche. More recently Bloemendal and Virag have shown this picture persists in the context of the general beta ensembles, giving new formulations of the critical limit laws . Yet another route, explained here, is to go through the random matrix hard edge, perturbing the smallest eigenvalues in the sample covariance set-up. A limiting procedure then recovers all the alluded to distributions. (Joint work with Jose Ramirez.)

Title: Extreme values for random processes of tree structures

Abstract: The main theme of this talk is that studying implicit tree structures of random processes is of significance in understanding their extreme values. I will illustrate this by several examples including cover times for random walks, maxima for two-dimensional discrete Gaussian free fields, and stochastic distance models. Our main results include (1) An approximation of the cover time on any graph up to a multiplicative constant by the maximum of the Gaussian free field, which yields a deterministic polynomial-time approximation algorithm for the cover time (D.-Lee-Peres 2010); the asymptotics for the cover time on a bounded-degree graph by the maximum of the GFF (D. 2011); a bound on the cover time fluctuations on the 2D lattice (D. 2011). (2) Exponential and doubly exponential tails for the maximum of the 2D GFF (D. 2011); some results on the extreme process of the 2D GFF (D.-Zeitouni, in preparation). (3) Critical and near-critical behavior for the mean-field stochastic distance model (D. 2011).

Title: Imaginary Geometry and the Gaussian Free Field

Abstract: The Schramm-Loewner evolution (SLE) is the canonical model of a non-crossing conformally invariant random curve, introduced by Oded Schramm in 1999 as a candidate for the scaling limit of loop erased random walk and the interfaces in critical percolation. The development of SLE has been one of the most exciting areas in probability theory over the last decade because Schramm's curves have now been shown to arise as the scaling limit of the interfaces of a number of different discrete models from statistical physics. In this talk, I will describe how SLE curves can be realized as the flow lines of a random vector field generated by the Gaussian free field, the two-time-dimensional analog of Brownian motion. I will also explain how this perspective can be used to prove several new results regarding the sample path behavior of SLE, in particular reversibility for kappa in (4,8). Based on joint works with Scott Sheffied.

Title: Directed random polymers and Macdonald processes

Abstract: The goal of the talk is to survey recent progress in understanding statistics of certain exactly solvable growth models, particle systems, directed polymers in one space dimension, and stochastic PDEs. A remarkable connection to representation theory and integrable systems is at the heart of Macdonald processes, which provide an overarching theory for this solvability. This is based off of joint work with Alexei Borodin.

Title: Path properties of the Schramm-Loewner Evolution.

Title: Extrema of branching Brownian motion

Abstract: Branching Brownian motion (BBM) on the real line is a particle system where particles perform Brownian motion and independently split into two independent Brownian particles after an exponential holding time. The statistics of extremal particles of BBM in the limit of large time are of interest for physicists and probabilists since BBM constitutes a borderline case, among Gaussian processes, where correlations affect the statistics. In this talk, I will start by reviewing results on the law of the maximum of BBM (the rightmost particle), and present new results on the joint distribution of particles close to the maximum. In particular, I will show how the approach can be used to prove ergodicity of the particle system. If time permits, I will explain how the program for BBM lays out a road map to understand extrema of log-correlated Gaussian fields such as the 2D Gaussian free field. This is joint work with A. Bovier and N. Kistler.

This is a special event. Billingsley Lectures on Probability in honor of Patrick Billingsley

Abstract: See it here.

Abstract: See it here.

Title: Landscape of random functions in many dimensions via Random Matrix Theory.

Abstract: How many critical values a typical Morse function have on a high dimensional manifold? Could we say anything about the topology of its level sets? In this talk I will survey a joint work with Gerard Ben Arous and Jiri Cerny that addresses these questions in a particular but fundamental example. We investigate the landscape of a general Gaussian random smooth function on the N-dimensional sphere. These corresponds to Hamiltonians of well-known models of statistical physics, i.e spherical spin glasses. Using the classical Kac-Rice formula, this counting boils down to a problem in Random Matrix Theory. This allows us to show an interesting picture for the complexity of these random Hamiltonians, for the bottom of the energy landscape, and in particular a strong correlation between the index and the critical value. We also propose a new invariant for the possible transition between the so-called 1-step replica symmetry breaking and a Full Replica symmetry breaking scheme and show how the complexity function is related to the Parisi functional.

Title: Finite-rank deformations of Wigner matrices.

Abstract: The spectral statistics of large Wigner matrices are by now well-understood. They exhibit the striking phenomenon of universality: under very general assumptions on the matrix entries, the limiting spectral statistics coincide with those of a Gaussian matrix ensemble. I shall talk about Wigner matrices that have been perturbed by a finite-rank matrix. By Weyl's interlacing inequalities, this perturbation does not affect the large-scale statistics of the spectrum. However, it may affect eigenvalues near the spectral edge, causing them to break free from the bulk spectrum. In a series of seminal papers, Baik, Ben Arous, and Peche (2005) and Peche (2006) established a sharp phase transition in the statistics of the extremal eigenvalues of perturbed Gaussian matrices. At the BBP transition, an eigenvalue detaches itself from the bulk and becomes an outlier. I shall report on recent joint work with Jun Yin. We consider an NxN Wigner matrix H perturbed by an arbitrary deterministic finite-rank matrix A. We allow the eigenvalues of A to depend on N. Under optimal (up to factors of log N) conditions on the eigenvalues of A, we identify the limiting distribution of the outliers. We also prove that the remaining eigenvalues "stick" to eigenvalues of H, thus establishing the edge universality of H + A. On the other hand, our results show that the distribution of the outliers is not universal, but depends on the distribution of H and on the geometry of the eigenvectors of A. As the outliers approach the bulk spectrum, this dependence is washed out and the distribution of the outliers becomes universal.

Jonathan Mattingly - Duke University

Title: A Menagerie of Stochastic Stabilization

Abstract: A basic problem for a stochastic system is to show that it possesses a unique steady state which dictates the long term statistics of the system. Sometimes the existence of such a measure is the difficult part. One needs control of the excursions away from the systems typical scale. As in deterministic system, one popular method is the construction of a Lyapunov Function. In the stochastic setting there lack of systematic methods to construct a Lyapunov Function when the interplay between the deterministic dynamics and stochastic dynamics are important for stabilization. I will give some modest steps in this direction which apply to a number of cases. In particular I will show a system where an explosive deterministic system is stabilized by the addition of noise and examples of physical systems where it is not clear how the deterministic system absorbs the stochastic excitation with out blowing up.

Title: From random interlacements to coordinate and infinite cylinder percolation

Abstract: During the talk I will focus on the connectivity properties of three models with long (infinite) range dependencies: Random Interlacements, percolation of the vacant set in infinite rod model and Coordinate percolation. The latter model have polynomial decay in sub-critical and super-critical regime in dimension 3. I will explain the nature of this phenomenon and why it is difficult to handle these models technically. In the second half of the talk I will present key ideas of the multi-scale analysis which allows to reach some conclusions. At the end I will discuss applications and several open problems.

Title: Complete matchings and random matrix theory

Abstract: Over the last decade or so, it has been found that the distributions that first appeared in random matrix theory describe several objects in probability and combinatorics which do not come from matrix at all. We consider one such example from the so-called maximal crossing and nesting of random complete matchings of integers. We also discuss related non-intersecting process. This is a joint work with Bob Jenkins.

Title: A simplified proof of the relation between scaling exponents in first-passage percolation

Abstract: In first passage percolation, we place i.i.d. non-negative weights on the nearest-neighbor edges of Z^d and study the induced random metric. A long-standing conjecture gives a relation between two "scaling exponents": one describes the variance of the distance between two points and the other describes the transversal fluctuations of optimizing paths between the same points. This is sometimes referred to as the "KPZ relation." In a recent breakthrough work, Sourav Chatterjee proved this conjecture using a strong definition of the exponents. I will discuss work I just completed with Tuca Auffinger, in which we introduce a new and intuitive idea that replaces Chatterjee's main argument and gives an alternative proof of the relation. One advantage of our argument is that it does not require a certain non-trivial technical assumption of Chatterjee on the weight distribution.

Ofer Zeitouni - University of Minnessota

Title: Traveling waves, branching random walks, and the Gaussian free field

Abstract: I will discuss several aspects of Branching random walks and their relation with the KPP equation on the one hand, and the maximum of certain (two dimensional) Gaussian fields on the other. I will not assume any knowledge about either of these terms.

Title: The parafermionic observable in Schramm-Loewner Evolutions

Abstract: In recent years, work by Stanislav Smirnov and his co-authors has greatly advanced our understanding of discrete stochastic processes, such as self-avoiding walk and the Ising model, via the use of a tool known as the parafermionic observable. Much of that work has been done in order to show convergence of these models to Schramm-Loewner Evolutions (SLE) in the scaling limit, although very little work has been done on what the parafermionic observable is in SLE itself. In this talk I will introduce the parafermionic observable, and then discuss one possible generalization to the continuous setting. I will then briefly introduce SLE and compute its parafermionic observable, ending with a couple of open questions.

Title: The contact process on the complete graph with random, vertex-dependent infection rates.

Abstract: The contact process is an interacting particle system that is a very simple model for the spread of an infection or disease on a network. Traditionally, the contact process was studied on homogeneous graphs such as the integer lattice or regular trees. However, due to the non-homogeneous structure of many real-world networks, there is currently interest in studying interacting particle systems in non-homogeneous graphs and environments. In this talk, I consider the contact process on the complete graph, where the vertices are assigned (random) weights and the infection rate between two vertices is proportional to the product of their weights. This set-up allows for some interesting analysis of the process and detailed calculations of phase transitions and critical exponents.

Title: Universality for beta-ensembles.

Abstract: Wigner stated the general hypothesis that the distribution of eigenvalue spacings of large complicated quantum systems is universal in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. The simplest case for this hypothesis is for ensembles of large but finite dimensional matrices. Spectacular progress was done in the past decade to prove universality of random matrices presenting an orthogonal, unitary or symplectic invariance. These models correspond to log-gases with respective inverse temperature 1, 2 or 4. I will report on a joint work with L. Erd\"os and H.-T. Yau, which yields universality for the log-gases at arbitrary temperature. The involved techniques include a multiscale analysis and a local logarithmic Sobolev inequality.

Friday, Oct. 8, Fredrik Johansson Viklund, Columbia U.

Friday, Oct. 15, Midwest Probability Colloquium at Northwestern

Friday, Oct. 29, Tom Alberts, U. of Toronto, Convergence of Loop-Erased Random Walk to SLE(2) in the Natural Time Parameterization

I will discuss work in progress with Michael Kozdron and Robert Masson on the convergence of the two-dimensional loop-erased random walk process to SLE(2), with the time parameterization of the curves taken into account. This is a strengthening of the original Lawler, Schramm, and Werner result which was only for curves modulo a reparameterization. The ultimate goal is to show that the limiting curve is SLE(2) with the very specific natural time parameterization that was recently introduced in Lawler and Sheffield, and further studied in Lawler and Zhou. I will describe several possible choices for the parameterization of the discrete curve that should all give the natural time parameterization in the limit, but with the key difference being that some of these discrete time parameterizations are easier to analyze than the others.

Friday, Dec. 3, Pierre Nolin, Courant Institute Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model

For two-dimensional independent percolation, Russo-Seymour-Welsh (RSW) bounds on crossing probabilities are an important a-priori indication of scale invariance, and they turned out to be a key tool to describe the phase transition: what happens at and near criticality. In this talk, we prove RSW-type uniform bounds on crossing probabilities for the FK Ising model at criticality, independent of the boundary conditions. A central tool in our proof is Smirnov's fermionic observable for the FK Ising model, that makes some harmonicity appear on the discrete level, providing precise estimates on boundary connection probabilities. We also prove several related results - including some new ones - among which the fact that there is no magnetization at criticality, tightness properties for the interfaces, and the value of the half-plane one-arm exponent. This is joint work with H. Duminil-Copin and C. Hongler.