Abstract. We study the class of monotone, two-state, deterministic cellular automata, in which sites are activated (or `infected') by certain congurations of nearby infected sites. These models have close connections to statistical physics, and several specific examples have been extensively studied in recent years by both mathematicians and physicists. This general setting was first studied only recently, however, by Bollobas, Smith and Uzzell, who showed that the family of all such `bootstrap percolation' models on $\mathbb Z^2$ can be naturally partitioned into three classes, which they termed subcritical, critical and supercritical.
In this paper we determine the order of the threshold for percolation (complete occupation) for every critical bootstrap percolation model in two dimensions. This `universality' theorem includes as special cases results of Aizenman and Lebowitz, Gravner and Grieath, Mountford, and van Enter and Hulshof, signicantly strengthens bounds of Bollobas, Smith and Uzzell, and complements recent work of Balister, Bollobas, Przykucki and Smith on subcritical models.

Abstract. We prove that the $q$-state Potts model and the random-cluster model with cluster weight $q>4$ undergo a discontinuous phase transition on the square lattice. More precisely, we show
1. Existence of multiple infinite-volume measures for the critical Potts and random-cluster models,
2. Ordering for the measures with monochromatic (resp. wired) boundary conditions for the critical Potts model (resp. random-cluster model), and
3. Exponential decay of correlations for the measure with free boundary conditions for both the critical Potts and random-cluster models. The proof is based on a rigorous computation of the Perron-Frobenius eigenvalues of the diagonal blocks of the transfer matrix of the six-vertex model, whose ratios are then related to the correlation length of the random-cluster model. As a byproduct, we rigorously compute the correlation lengths of the critical random-cluster and Potts models, and show that they behave as $\exp(\pi^2/\sqrt{q-4})$ as~$q$ tends to 4.

Abstract. The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been predicted by Nienhuis that for $0\le n\le 2$ the loop $O(n)$ model exhibits a phase transition at a critical parameter $x_c(n)=1/\sqrt{2+\sqrt{2-n}}$. For $0 < n\le 2$, the transition line has been further conjectured to separate a regime with short loops when $x < x_c(n)$ from a regime with macroscopic loops when $x\ge x_c(n)$. In this paper, we prove that for $n\in [1,2]$ and $x=x_c(n)$ the loop $O(n)$ model exhibits macroscopic loops. This is the first instance in which a loop $O(n)$ model with $n\neq 1$ is shown to exhibit such behaviour. A main tool in the proof is a new positive association (FKG) property shown to hold when $n \ge 1$ and $0 < x\le 1/\sqrt{n}$. This property implies, using techniques recently developed for the random-cluster model, the following dichotomy: either long loops are exponentially unlikely or the origin is surrounded by loops at any scale (box-crossing property). We develop a ``domain gluing'' technique which allows us to employ Smirnov's parafermionic observable to rule out the first alternative when $x=x_c(n)$ and $n\in[1,2]$.

Abstract. We relate the planar random current representation introduced by Griffiths, Hurst and Sherman to the dimer model. More precisely, we provide a measure-preserving map between double random currents (obtained as the sum of two independent random currents) on a planar graph and dimers on an associated bipartite graph. We also construct a nesting field for the double random current, which, under this map, corresponds to the height function of the dimer model. As applications, we provide an alternative derivation of some of the bozonization rules obtained recently by Dub\'edat, and show that the spontaneous magnetization of the Ising model on a planar biperiodic graph vanishes at criticality.

Abstract. We prove that for Voronoi percolation on $\mathbb R^d$, there exists $p_c\in[0,1]$ such that

- for $p$ smaller than $p_c$, there exists $c_p>0$ such that $\mathbb P_p[0\text{ connected to distance }n]\le \exp(-c_pn)$,
- there exists $c>0$ such that for $p>p_c$, $\mathbb P_p[0\text{ connected to }\infty]\ge c(p-p_c)$.

This result offers a new way of showing that $p_c=1/2$, and the first proof of mean-field lower bound for the density of the infinite cluster, even in dimension 2. This paper belongs to a series of papers using the theory of algorithms to prove sharpness of the phase transition.

Abstract. We study the critical probability for the metastable phase transition of the two-dimensional anisotropic bootstrap percolation model with $(1,2)$-neighbourhood and threshold $r = 3$. The first order asymptotics for the critical probability were recently determined by the first and second authors. Here we determine the following sharp second and third order asymptotics: \[
p_c\big( [L]^2,\mathcal{N}_{(1,2)},3 \big) \; = \; \frac{(\log \log L)^2}{12\log L} \, - \, \frac{\log \log L \, \log \log \log L}{ 3\log L}
+ \frac{\left(\log \frac{9}{2} + 1 \pm o(1) \right)\log \log L}{6\log L}. \] We note that the second and third order terms are so large that the first order asymptotics fail to approximate $p_c$ even for lattices of size well beyond $10^{10^{1000}}$.

Abstract. We prove an inequality on decision trees on monotonic measures which generalizes the OSSS inequality on product spaces. As an application, we use this inequality to prove a number of new results on lattice spin models and their random-cluster representations. More precisely, we prove that
- For the Potts model on transitive graphs, correlations decay exponentially fast below criticality.
- For the random-cluster model with cluster weight $q\ge1$ on transitive graphs, correlations decay exponentially fast in the subcritical regime and the cluster-density satisfies the mean-field lower bound in the supercritical regime.
- For the random-cluster models with cluster weight $q\ge1$ on planar quasi-transitive graphs $\mathbb G$,
$\frac{p_c(\mathbb G)p_c(\mathbb G^*)}{(1-p_c(\mathbb G))(1-p_c(\mathbb G^*))}~=~q.$
As a special case, we obtain the value of the critical point for the square, triangular and hexagonal lattices (this provides a short proof of the result of Beffara and Duminil-Copin. These results have many applications for the understanding of the subcritical (respectively disordered) phase of all these models. The techniques developed in this paper have potential to be extended to a wide class of models including the Ashkin-Teller model, continuum percolation models such as Voronoi percolation and Boolean percolation, super-level sets of massive Gaussian Free Field, and random-cluster and Potts model with infinite range interactions.

Abstract. We show that the canonical random-cluster measure associated to isoradial graphs is critical for all $q \geq 1$. Additionally, we prove that the phase transition of the model is of the same type on all isoradial graphs: continuous for $1 \leq q \leq 4$ and discontinuous for $q > 4$. For $1 \leq q \leq 4$, the arm exponents (assuming their existence) are shown to be the same for all isoradial graphs. In particular, these properties also hold on the triangular and hexagonal lattices. Our results also include the limiting case of quantum random-cluster models in $1+1$ dimensions.

Abstract. The class of critical bootstrap percolation models in two dimensions was recently introduced by Bollobás, Smith and Uzzell, and the critical threshold for percolation was determined up to a constant factor for all such models by the authors of this paper. Here we develop and refine the techniques introduced in that paper in order to determine a sharp threshold for the Duarte model. This resolves a question of Mountford from 1995, and is the first result of its type for a model with drift.

Abstract. The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been conjectured that both the spin and the loop $O(n)$ models exhibit exponential decay of correlations when $n>2$. We verify this for the loop $O(n)$ model with large parameter $n$, showing that long loops are exponentially unlikely to occur, uniformly in the edge weight $x$. Our proof provides further detail on the structure of typical configurations in this regime. Putting appropriate boundary conditions, when $nx^6$ is sufficiently small, the model is in a dilute, disordered phase in which each vertex is unlikely to be surrounded by any loops, whereas when $nx^6$ is sufficiently large, the model is in a dense, ordered phase which is a small perturbation of one of the three ground states.

Abstract. This article studies the planar Potts model and its random-cluster representation. We show that the phase transition of the nearest-neighbor ferromagnetic $q$-state Potts model on $\mathbb Z^2$ is continuous for $q\in\{2,3,4\}$, in the sense that there exists a unique Gibbs state, or equivalently that there is no ordering for the critical Gibbs states with monochromatic boundary conditions. The proof uses the random-cluster model with cluster-weight $q\ge1$ (note that $q$ is not necessarily an integer) and is based on two ingredients: The fact that two-point function for the free state decays sub-exponentially fast for cluster-weights $1\le q\le 4$, which is derived studying parafermionic observables on a discrete Riemann surface. And a new result proving the equivalence of several properties of critical random-cluster models: - the absence of infinite-cluster for wired boundary conditions,
- the uniqueness of infinite-volume measures,
- the sub-exponential decay of the two-point function for free boundary conditions,
- a Russo-Seymour-Welsh type result on crossing probabilities in rectangles with {\em arbitrary boundary conditions}.
The result leads to a number of consequences concerning the scaling limit of the random-cluster model with $q\in[1,4]$. It shows that the family of interfaces (for instance for Dobrushin boundary conditions) are tight when taking the scaling limit and that any sub-sequential limit can be parametrized by a Loewner chain. We also study the effect of boundary conditions on these sub-sequential limits. Let us mention that the result should be instrumental in the study of critical exponents as well.

Abstract. We consider translationally-invariant percolation models on $\mathbb Z^d$ satisfying the finite energy and the FKG properties. We provide explicit upper bounds on the probability of having two distinct clusters going from the endpoints of an edge to distance $n$ (this corresponds to a finite size version of the celebrated Burton-Keane argument proving uniqueness of the infinite-cluster). The proof is based on the generalization of a reverse Poincaré inequality proved by Sen and Chatterjee. As a consequence, we obtain upper bounds on the probability of the so-called four-arm event for planar random-cluster models with cluster-weight $q\geq1$.

Abstract. We prove that random-cluster models with $q \geq 1$ on a variety of planar lattices have a sharp phase transition, that is that there exists some parameter $p_c$ below which the model exhibits exponential decay and above which there exists a.s. an infinite cluster. The result may be extended to the Potts model via the Edwards-Sokal coupling. Our method is based on sharp threshold techniques and certain symmetries of the lattice; in particular it makes no use of self-duality. Due to its nature, this strategy could be useful in studying other planar models satisfying the FKG lattice condition and some additional differential inequalities.

Abstract. We study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on $\mathbb Z^d$. We prove that the vector space of harmonic functions growing at most linearly is $d+1$-dimensional almost surely. In particular, there are no non-constant sublinear harmonic functions (thus implying the uniqueness of the corrector). The main ingredient of the proof is given by a quantitative, annealed version of the Kaimanovich-Vershik entropy argument. This also provides bounds on the derivative of the heat kernel, simplifying and generalizing existing results. Even reversibility is not necessary. We also mention several open problems and conjectures on the behavior of harmonic functions on stationary random graphs.

Abstract. The spontaneous magnetization is proved to vanish continuously at the critical temperature for a class of ferromagnetic Ising spin systems which includes the nearest neighbor ferromagnetic Ising spin model on $\mathbb Z^d$ in $d=3$ dimensions. The analysis applies also to higher dimensions, for which the result is already known, and to systems with interactions of power law decay. The proof employs in an essential way an extension of Ising model's random current representation to the model's infinite volume limit. Using it, we relate the continuity of the magnetization to the vanishing of the free boundary condition Gibbs state's Long Range Order parameter. For reflection positive models the resulting criterion for continuity may be established through the infrared bound for all but the borderline case, of the one dimensional model with $1/r^2$ interaction, for which the spontaneous magnetization is known to be discontinuous at $T_c$.

Abstract. The critical surface for random-cluster model with cluster-weight $q\ge 4$ on isoradial graphs is identified using parafermionic observables. Correlations are also shown to decay exponentially fast in the subcritical regime. While this result is restricted to random-cluster models with $q\ge 4$, it extends the recent theorem of [BD12] to a large class of planar graphs. In particular, the anisotropic random-cluster model on the square lattice are shown to be critical if $\frac{p_vp_h}{(1-p_v)(1-p_h)}=q$, where $p_v$ and $p_h$ denote the horizontal and vertical edge-weights respectively. We also provide consequences for Potts models.

Comment. This article is part of the special issue Exactly solved models and beyond in honour of R. Baxter's 75th birthday.

Abstract. In 2010, the growth constant of self-avoiding walks on the hexagonal (a.k.a honeycomb) lattice was proved to be equal to $\sqrt{2+\sqrt 2}$: A key identity used in that proof was later generalised by Smirnov so as to apply to a general $O(n)$ loop model with $n\in[-2,2]$ (the case $n = 0$ corresponding to self-avoiding walks). We modify this model by restricting to a half-plane and introducing a fugacity associated with boundary sites (also called surface sites) and obtain a generalisation of Smirnov’s identity. The value of the critical surface fugacity was conjectured by Batchelor and Yung in 1995. This value also plays a crucial role in our identity, which thus provides an independent prediction for it. For the case $n = 0$, corresponding to self-avoiding walks interacting with a surface, we prove the conjectured value of the critical surface fugacity. A critical part of this proof involves demonstrating that the generating function of self-avoiding bridges of height $T$, taken at its critical point, tends to 0 as $T$ increases, as predicted from SLE theory.

Abstract. We prove that all Gibbs states of the q-state nearest neighbor Potts model on $\mathbb Z^2$ below the critical temperature are convex combinations of the $q$ pure phases; in particular, they are all translation invariant. To achieve this goal, we consider such models in large finite boxes with arbitrary boundary condition, and prove that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the inverse temperature $\beta \ge\beta_c$.


© Vincent Beffara

Abstract. Internal Diffusion Limited Aggregation (IDLA) is a model that describes the growth of a random aggregate of particles from the inside out. Shellef proved that IDLA processes on supercritical percolation clusters of integer-lattices fill Euclidean balls, with high probability. In this article, we complete the picture and prove a limit-shape theorem for IDLA on such percolation clusters, by providing the corresponding upper bound. The technique to prove upper bounds is new and robust: it only requires the existence of a good lower bound. Specifically, this way of proving upper bounds on IDLA clusters is more suitable for random environments than previous ways, since it does not harness harmonic measure estimates.

© Cyrille Lucas

Abstract. Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study an anisotropic bootstrap percolation model. We prove that it exhibits a sharp metastability threshold. This is the first mathematical proof of a sharp threshold for an anisotropic bootstrap percolation model.

Erratum. Although Theorem 1.1 as stated in our paper is correct, the generalisation announced at the end of page 1219 and the beginning of page 1220 to the situation where the site $(m,n)$ gets occupied if $k+1$ sites among the $2k+2$ sites $(m+1,n)$, $(m,n-1)$ and $(m-k,n),\dots, (m-1,n), (m+1,n),\dots,(m+k,n)$ are occupied, is incorrect as stated for $k\ge2$. It should read that (in probability) $\frac{1}{p}(\log \frac{1}{p})^2 \log T$ tends to $\frac{(k-1)^2}{4(k+1)}$. In other words, the constant $\frac{1}{4(k+1)}$ given in our paper on the right-hand side of the unnumbered equation on page 1220 should be replaced by $\frac{(k-1)^2}{4(k+1)}$. We thank Rob Morris for pointing this out to us.
Also, the definition of weakly connected in page 1226 should be modified as follows. Two occupied points $x,y\in\mathbb Z^2$ are weakly connected if $x\in y+\mathcal N$ or there exists $z\in \mathbb Z^2$ such that $x,y\in z+\mathcal N$. We thank Tim Hulshof for pointing this out to us.

Abstract. We provide the first mathematical proof that the connective constant of the hexagonal lattice is equal to $\sqrt{2+\sqrt 2}$. This value has been derived non rigorously by B. Nienhuis in 1982, using Coulomb gas approach from theoretical physics. Our proof uses a parafermionic observable for the self avoiding walk, which satisfies a half of the discrete Cauchy-Riemann relations. Establishing the other half of the relations (which conjecturally holds in the scaling limit) would also imply convergence of the self-avoiding walk to SLE(8/3).

© Vincent Beffara

Abstract. Parafermionic observables were introduced by Smirnov for planar FK percolation in order to study the critical phase $(p; q) = (p_c(q); q)$. This article gathers several known properties of these observables. Some of these properties are used to prove the divergence of the correlation length when approaching the critical point for FK percolation when $1\le q\le 4$. A crucial step is to consider FK percolation on the universal cover of the punctured plane. We also mention several conjectures on FK percolation with arbitrary cluster-weight $q \ge 0$.

Comment. This article is part of 'Lattice models and integrability', a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.

Abstract. We prove a long-standing conjecture on random-cluster models, namely that the critical point for such models with parameter $q\ge1$ on the square lattice is equal to the self-dual point $p_{\rm sd}(q)=\sqrt{q}/(1+\sqrt{q})$. This gives a rigorous proof that the critical temperature of the q-state Potts model is equal to $\log(1+\sqrt q)$ for all $q\ge 1$. We further prove that the transition is sharp, meaning that there is exponential decay of correlations in the sub-critical phase. The techniques of this paper are rigorous and valid for all $q\ge1$, in contrast to earlier methods valid only for certain given $q$. The proof extends to the triangular and the hexagonal lattices as well.

Abstract. In $r$-neighbour bootstrap percolation on a graph $G$, a (typically random) set $A$ of initially 'infected' vertices spreads by infecting (at each time step) vertices with at least $r$ already-infected neighbours. This process may be viewed as a monotone version of the Glauber dynamics of the Ising model, and has been extensively studied on the $d$-dimensional grid $[n]^d$. The elements of the set $A$ are usually chosen independently, with some density $p$, and the main question is to determine $p_c([n]^d,r)$, the density at which percolation (infection of the entire vertex set) becomes likely.
In this paper we prove, for every pair $d \ge r \ge 2$, that there is a constant $L(d,r)$ such that $p_c([n]^d,r) = [(L(d,r) + o(1)) / log_(r-1) (n)]^{d-r+1}$ as $n \to \infty$, where $log_r$ denotes an $r$-times iterated logarithm. We thus prove the existence of a sharp threshold for percolation in any (fixed) number of dimensions. Moreover, we determine $L(d,r)$ for every pair $(d,r)$.

Abstract. Motivated by Kesten’s bridge decomposition for two-dimensional self-avoiding walks in the upper half plane, we show that the conjectured scaling limit of the half-plane SAW, the SLE(8/3) process, also has an appropriately defined bridge decomposition. This continuum decomposition turns out to entirely be a consequence of the restriction property of SLE(8/3), and as a result can be generalized to the wider class of restriction measures. Specifically we show that the restriction hulls with index less than one can be decomposed into a Poisson Point Process of irreducible bridges in a way that is similar to Ito’s excursion decomposition of a Brownian motion according to its excursions.

© Tom Alberts

A precision: In the definition p330 of $\mathcal E_\Omega(c,d)$, the interface from $c$ to $d$ is required to be edge-avoiding with other loops, and the number of loops includes the interface.

Erratum: Mistake in the formula of $\delta_x$ in (6.4) coming from a mistake in the second displayed equation in p160. See the original article for the relevant modification.

Abstract. These lecture notes describe the content of a six-hours course given by the two authors at the 2012 probability summer school in Saint-Petersburg. The goal is to provide a derivation of several critical parameters of classical planar models such as Bernoulli and Fortuin-Kasteleyn percolation as well as the Ising and Potts models.

Comment. Course given jointly with V. Beffara at the St Petersburg summer School 2012.

Abstract. Critical phenomena at an order/disorder phase transition has been a central object of interest in statistical physics. In the past century, techniques borrowed from many different fields of mathematics (Algebra, Combinatorics, Probability, Complex Analysis, Spectral Theory, etc) have contributed to a more and more elaborated description of the possible critical behaviors for a large variety of models (interacting particle systems, lattice spin models, spin glasses, percolation models). Through the classical examples of the Ising and Potts models, we survey a few recent advances regarding the rigorous understanding of such phase transitions for the specific case of lattice spin models. This review was written at the occasion of the Harvard/MIT conference Current Developments in Mathematics 2015.


© Vincent Beffara

Abstract.We provide a proof that the random-cluster model on the square lattice undergoes a discontinuous phase transition for large values of the cluster-weight $q$. This implies discontinuity of the phase transition for Potts model on the square lattice provided that the number of colors $q$ is large enough. Let us remind the reader that this result is classical and that we simply provide an alternative approach based on the loop representation.