By the FKG inequality for FK-percolation, the probability of the alternating twoarm event is smaller than the product of the probabilities of having a primal arm and a dual arm, respectively. In this paper, we improve this inequality by a polynomial factor for critical planar FK-percolation in the continuous phase transition regime (1 ≤ q ≤ 4). In particular, we prove that if the alternating two-arm exponent $\alpha_{01}$ α₀₁ and the one-arm exponents $\alpha_{1}$ and $\alpha_{1}$ exist, then they satisfy the strict inequality $\alpha_{01}>\alpha_{0}+\alpha_{1}$.
The question was formulated in by Garban and Steif (2014, Open problem XIII.6) in the context of exceptional times and was brought to our attention by Ritvik Ramanan Radhakrishnan and Vincent Tassion, who obtained the same result for planar Bernoulli percolation through different methods.

We answer the following question: if the occupied (or vacant) set of a planar Poisson Boolean percolation model does contain a crossing of an $n x n$ square, how wide is this crossing? The answer depends on the whether we consider the critical, sub- or supercritical regime, and is different for the occupied and vacant sets.

We provide a new proof of the near-critical scaling relation \(\beta=\xi_1\nu\) for Bernoulli percolation on the square lattice already proved by Kesten in 1987. We rely on a novel approach that does not invoke Russo's formula, but rather relates differences in crossing probabilities at different scales. The argument is shorter and more robust than previous ones and is more likely to be adapted to other models. The same approach may be used to prove the other scaling relations appearing in Kesten's work.

Abstract. We prove that all Gibbs measures of the q-state Potts model on the 2D square lattice are linear combinations of the extremal measures obtained as thermodynamic limits under free or monochromatic boundary conditions. In particular all Gibbs measures are invariant under translations. This statement is new at points of first-order phase transition, that is at \(T = T_c(q)\) when \(q > 4\). In this case the structure of Gibbs measures is the most complex in the sense that there exist q + 1 distinct extremal measures
Most of the work is devoted to the FK-percolation model on \(Z^2\) with \(q \geq 1\), where we prove that every Gibbs measure is a linear combination of the free and wired ones. The arguments are non-quantitative and follow the spirit of the seminal works of Aizenman and Higuchi, which established the Gibbs structure for the two-dimensional Ising model. Infinite-range dependencies in FK-percolation (i.e., a weaker spatial Markov property) pose serious additional difficulties compared to the case of the Ising model. For example, it is not automatic, albeit true, that thermodynamic limits are Gibbs. The result for the Potts model is then derived using the Edwards–Sokal coupling and auto-duality. The latter ingredient is necessary since applying the Edwards–Sokal procedure to a Gibbs measure for the Potts model does not automatically produce a Gibbs measure for FK-percolation.
Finally, the proof is generic enough to adapt to the FK-percolation and Potts models on the triangular and hexagonal lattices and to the loop O(n) model in the range of parameters for which its spin representation is positively associated.

Abstract. We show that the height function of the six-vertex model, in the parameter range \(\mathbf a=\mathbf b=1\) and \(\mathbf c\ge1\), is delocalized with logarithmic variance when \(\mathbf c\le 2\). This complements the earlier proven localization for \(\mathbf c>2\). Our proof relies on Russo-Seymour-Welsh type arguments, and on the local behaviour of the free energy of the cylindrical six-vertex model, as a function of the unbalance between the number of up and down arrows.

Abstract. In this paper, we prove that the large scale properties of a number of two-dimensional lattice models are rotationally invariant. More precisely, we prove that the random-cluster model on the square lattice with cluster-weight \(1\le q\le 4\) exhibits rotational invariance at large scales. This covers the case of Bernoulli percolation on the square lattice as an important example. We deduce from this result that the correlations of the Potts models with \(q\in\{2,3,4\}\) colors and of the six-vertex height function with \(\Delta\in[-1,-1/2]\) are rotationally invariant at large scales.

Abstract. In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime \(\Delta<1\). As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches 1/2. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when \(a=b=1\) and \(c\ge1\), and the rotational invariance of the six-vertex model and the Fortuin-Kasteleyn percolation.

Abstract. This paper studies the critical and near-critical regimes of the planar random-cluster model on \(\mathbb Z^2\) with cluster-weight \(q\in[1,4]\) using novel coupling techniques. More precisely, we derive the scaling relations between the critical exponents \(\beta, \gamma, \delta, \eta, \nu, \zeta\) as well as \(\alpha\) (when \(\alpha\ge0\)). As a key input, we show the stability of crossing probabilities in the near-critical regime using new interpretations of the notion of influence of an edge in terms of the rate of mixing. As a byproduct, we derive a generalization of Kesten's classical scaling relation for Bernoulli percolation involving the ``mixing rate'' critical exponent \(\iota\) replacing the four-arm event exponent \(\xi_4\).

Abstract. This paper is studying the critical regime of the planar random-cluster model on \(\mathbb Z^2\) with cluster-weight \(q\in[1,4)\). More precisely, we prove crossing estimates in quads which are uniform in their boundary conditions and depend only on their extremal lengths. They imply in particular that any fractal boundary is touched by macroscopic clusters, uniformly in its roughness or the configuration on said boundary. Additionally, they imply that any sub-sequential scaling limit of the collection of interfaces between primal and dual clusters is made of loops that are non-simple.
We also obtain a number of properties of so-called arm-events: three universal critical exponents (two arms in the half-plane, three arms in the half-plane and five arms in the bulk), quasi-multiplicativity and well-separation properties (even when arms are not alternating between primal and dual), and the fact that the four-arm exponent is strictly smaller than 2. These results were previously known only for Bernoulli percolation (\(q = 1\)) and the FK-Ising model (\(q = 2\)).
Finally, we prove new bounds on the one, two and four arms exponents for \(q\in[1,2]\). These improve the previously known bounds, even for Bernoulli percolation.

Abstract. We study randomly growing trees governed by the affine preferential attachment rule. Starting with a seed tree \(S\), vertices are attached one by one, each linked by an edge to a random vertex of the current tree, chosen with a probability proportional to an affine function of its degree. This yields a one-parameter family of preferential attachment trees \((T_n^S)_{n\geq|S|}\), of which the linear model is a particular case. Depending on the choice of the parameter, the power-laws governing the degrees in \(T_n^S\) have different exponents.
We study the problem of the asymptotic influence of the seed \(S\) on the law of \(T_n^S\). We show that, for any two distinct seeds \(S\) and \(S'\), the laws of \(T_n^S\) and \(T_n^{S'}\) remain at uniformly positive total-variation distance as \(n\) increases.
This is a continuation of Curien et al. (2015), which in turn was inspired by a conjecture of Bubeck et al. (2015). The technique developed here is more robust than previous ones and is likely to help in the study of more general attachment mechanisms.

Abstract. We show that the loop \(O(n)\) model exhibits exponential decay of loop sizes whenever \(n\geq 1\) and \(x<\tfrac{1}{\sqrt{3}}+\varepsilon(n)\), for some suitable choice of \(\varepsilon(n)>0\).
It is expected that, for \(n \leq 2\), the model exhibits a phase transition in terms of \(x\), that separates regimes of polynomial and exponential decay of loop sizes. In this paradigm, our result implies that the phase transition for \(n \in (1,2]\) occurs at some critical parameter \(x_c(n)\) strictly greater than that \(x_c(1) = 1/\sqrt3\). The value of the latter is known since the loop \(O(1)\) model on the hexagonal lattice represents the contours of spin-clusters of the Ising model on the triangular lattice.
The proof is based on developing \(n\) as \(1+(n-1)\) and exploiting the fact that, when \(x<\tfrac{1}{\sqrt{3}}\), the Ising model exhibits exponential decay on any (possibly non simply-connected) domain. The latter follows from the positive association of the FK-Ising representation.

Abstract. Uniform integer-valued Lipschitz functions on a domain of size \(N\) of the triangular lattice are shown to have variations of order \(\sqrt{\log N}\).
The level lines of such functions form a loop \(O(2)\) model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop \(O(2)\) model is constructed as a thermodynamic limit and is shown to be unique. It contains only finite loops and has properties indicative of scale-invariance: macroscopic loops appearing at every scale. The existence of the infinite-volume measure carries over to height functions pinned at 0; the uniqueness of the Gibbs measure does not.
The proof is based on a representation of the loop \(O(2)\) model via a pair of spin configurations that are shown to satisfy the FKG inequality. We prove RSW-type estimates for a certain connectivity notion in the aforementioned spin model.

Abstract. Let \(c_n = c_n(d)\) denote the number of self-avoiding walks of length \(n\) starting at the origin in the Euclidean nearest-neighbour lattice \(\mathbb Z^d\). Let \(\mu = \lim_n c_n^{1/n}\) denote the connective constant of \(\mathbb Z^d\). In 1962, Hammersley and Welsh proved that, for each \(d \geq 2\), there exists a constant \(C > 0\) such that \(c_n \leq \exp(C n^{1/2}) \mu^n\) for all \(n \in \mathbb N\). While it is anticipated that \(c_n \mu^{-n}\) has a power-law growth in \(n\), the best known upper bound in dimension two has remained of the form \(n^{1/2}\) inside the exponential.
We consider two planar lattices and prove that \(c_n \leq \exp(C n^{1/2 -\epsilon}) \mu^n\) for an explicit constant \(\epsilon> 0\) (where here \(\mu\) denotes the connective constant for the lattice in question). The result is conditional on a lower bound on the number of self-avoiding polygons of length \(n\), which is proved to hold on the hexagonal lattice \(\mathbb H\) for all \(n\), and subsequentially in \(n\) for \(\mathbb Z^2\). A power-law upper bound on \(c_n \mu^{-n}\) for \(\mathbb H\) is also proved, contingent on a non-quantitative assertion concerning this lattice's connective constant.

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. We consider a self-avoiding walk model (SAW) on the faces of the square lattice \(\mathbb{Z}^2\). This walk can traverse the same face twice, but crosses any edge at most once. The weight of a walk is a product of local weights: each square visited by the walk yields a weight that depends on the way the walk passes through it. The local weights are parametrised by angles \(\theta\in[\frac{\pi}{3},\frac{2\pi}{3}]\) and satisfy the Yang--Baxter equation. The self-avoiding walk is embedded in the plane by replacing the square faces of the grid with rhombi with corresponding angles. By means of the Yang-Baxter transformation, we show that the 2-point function of the walk in the half-plane does not depend on the rhombic tiling (i.e. on the angles chosen). In particular, this statistic coincides with that of the self-avoiding walk on the hexagonal lattice. Indeed, the latter can be obtained by choosing all angles \(\theta\) equal to \(\frac{\pi}{3}\). For the hexagonal lattice, the critical fugacity of SAW was recently proved to be equal to \(1+\sqrt{2}\). We show that the same is true for any choice of angles. In doing so, we also give a new short proof to the fact that the partition function of self-avoiding bridges in a strip of the hexagonal lattice tends to 0 as the width of the strip tends to infinity. This proof also yields a quantitative bound on the convergence.

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
- Existence of multiple infinite-volume measures for the critical Potts and random-cluster models,
- Ordering for the measures with monochromatic (resp. wired) boundary conditions for the critical Potts model (resp. random-cluster model), and
- 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. In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector \(\psi\) and energy \(\Lambda\), which satisfy \(V \psi = \Lambda \psi\), where \(V\) is the the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights \(a= b=1\) and \(c > 0\). We also show that the same vector \(\psi\) satisfies \(H \psi = E \psi\), where \(H\) is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value \(E\) computed explicitly.
Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors that amounts to proving that the random cluster model on \(\mathbb{Z}^2\) with cluster weight \(q >4\) exhibits a first-order phase transition.

Abstract. For \(G\) a finitely generated group and \(g \in G\), we say \(g\) is detected by a normal subgroup \(N \lhd G\) if \(g \notin N\). The depth \(D_G(g)\) of \(g\) is the lowest index of a normal, finite index subgroup \(N\) that detects \(g\). In this paper we study the expected depth, \(\mathbb{E}[D_G(X_n)]\), where \(X_n\) is a random walk on \(G\). We give several criteria that imply that \[ \mathbb{E}[D_G(X_n)] \xrightarrow[n\to \infty]{} 2 + \sum_{k \geq 2}\frac{1}{[G:\Lambda_k]}\,,\] where \(\Lambda_k\) is the intersection of all normal subgroups of index at most \(k\). We explain how the right-hand side above appears as a natural limit and also give an example where the convergence does not hold.

Abstract. We prove sharpness of the phase transition for the random-cluster model with \(q \geq 1\) on graphs of the form \( S := G \times S \), where \(G\) is a planar lattice with mild symmetry assumptions, and \(S\) a finite graph. That is, for any such graph and any \(q \geq 1\), there exists some parameter \(p_c = p_c(S,q)\), below which the model exhibits exponential decay and above which there exists a.s. an infinite cluster.
The proof adapts the methods of [6] to non-planar graphs using techniques developed in [7]. The result is also valid for the random-cluster model on planar graphs with long range, compactly supported interaction. It extends to the Potts model via the Edwards- Sokal coupling.

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 pc 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. Part of the argument is not restricted to planar models and may be of some interest for the understanding of random-cluster and Potts models in higher dimensions.
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 are interested in the asymptotics of random trees built by linear preferential attachment, also known in the literature as Barabasi-Albert trees or plane-oriented recursive trees. We first prove a conjecture of Bubeck, Mossel & Racz concerning the influence of the seed graph on the asymptotic behavior of such trees. Separately we study the geometric structure of nodes of large degrees in a plane version of Barabasi-Albert trees via their associated looptrees. As the number of nodes grows, we show that these looptrees, appropriately rescaled, converge in the Gromov-Hausdorff sense towards a random compact metric space which we call the Brownian looptree. The latter is constructed as a quotient space of Aldous' Brownian Continuum Random Tree and is shown to have almost sure Hausdorff dimension 2.

Abstract. Self-destructive percolation with parameters \(p,\delta\) is obtained by taking a site percolation configuration with parameter \(p\), closing all sites belonging to infinite clusters, then opening every closed site with probability \(\delta\), independently of the rest. Call \(\theta(p,\delta)\) the probability that the origin is in an infinite cluster in the configuration thus obtained.
For two dimensional lattices, we show the existence of \(\delta > 0\)such that, for any \(p > p_c\), \(\theta(p,\delta) = 0\). This proves the conjecture of van den Berg and Brouwer, who introduced the model. Our results combined with those of van den Berg and Brouwer imply the non-existence of the infinite parameter forest-fire model. The methods herein apply to site and bond percolation on any two dimensional planar lattice with sufficient symmetry.

Abstract. We prove two results on the delocalization of the endpoint of a uniform self-avoiding walk on \(\mathbb{Z}^d\) for \(d \geq 2\). We show that the probability that a walk of length \(n\) ends at a point \(x\) tends to \(0\) as \(n\) tends to infinity, uniformly in \(x\). Also, when \(x\) is fixed, with \(\vert\vert x \vert\vert = 1\), this probability decreases faster than \(n^{-1/4 + \epsilon}\) for any positive \(\epsilon\).This provides a bound on the probability that a self-avoiding walk is a polygon.

Abstract. In an investigation of percolation on isoradial graphs, we prove the criticality of canonical bond percolation on isoradial embeddings of planar graphs, thus extending celebrated earlier results for homogeneous and inhomogeneous square, triangular, and other lattices. This is achieved via the star-triangle trasformation, by transporting the box-crossing property across the family of isoradial graphs. As a consequence, we obtain the universality of these models at the critical point, in the sense that the one-arm and 2j-alternating-arm critical exponents (and therefore also the connectivity and volume exponents) are constant across the family of such percolation processes. The isoradial graphs in question are those that satisfy certain weak conditions on their embedding and on their track system. This class of graphs includes, for example, isoradial embeddings of periodic graphs, and graphs derived from rhombic Penrose tilings.

Abstract. All (in)homogeneous bond percolation models on the square, triangular, and hexagonal lattices belong to the same universality class, in the sense that they have identical critical exponents at the critical point (assuming the exponents exist). This is proved using the star-triangle transformation and the box-crossing property. The exponents in question are the one-arm exponent \(\rho\), the \(2j\)-alternating-arms exponents \(\rho_{2j}\) for \(j \ge 1\), the volume exponent \(\deltalta\), and the connectivity exponent \(\eta\). By earlier results of Kesten, this implies universality also for the near-critical exponents \(\beta\), \(\gamma\), \(\nu\), \(\deltalta\) (assuming these exist) for any of these models that satisfy a certain additional hypothesis, such as the homogeneous bond percolation models on these three lattices.

Abstract. The star-triangle transformation is used to obtain an equivalence extending over the set of all (in)homogeneous bond percolation models on the square, triangular, and hexagonal lattices. Amongst the consequences are box-crossing (RSW) inequalities for such models with parameter-values at which the transformation is valid. This is a step towards proving the universality and conformality of these processes. It implies criticality of such values, thereby providing a new proof of the critical point of inhomogeneous systems. The proofs extend to certain isoradial models to which previous methods do not apply.