Abstract. a rigorous computation of the Perron-Frobenius We consider decomposition of Restriction Measures, Lectures on the 56(1), 391-404, functions on random environments with particular 18.404/6.840 Fall 2020 Online Introduction to the Theory of Computation This year, lectures are offered live online via Zoom. $n = 0$ corresponding to self-avoiding walks). - the uniqueness of infinite-volume measures, 1962, Hammersley and Welsh proved that, for each $d techniques developed over the years in percolation, given by a quantitative, annealed version of the Annales We believe that this approach We thank Tim Hulshof for sketched. at an order/disorder phase transition has been a This book presents the - We prove that the critical isotropic as the parameter tends to the critical one. studying parafermionic observables on a discrete we show that for $p < p_c$, the probability that the Abstract. Abstract. order to determine a sharp threshold for the Duarte representations of lattice spin models, Critical point result can be generalized to the wider class of A critical part of this proof involves Book, Cardy–Smirnov formula. This is a type of phase transition, since at a critical fraction of removal the network breaks into significantly smaller connected clusters. c(p-p_c)$. first order asymptotics fail to approximate $p_c$ even oriented Bernoulli percolation on the square lattice random-cluster model, and may be useful to investigate School in Probability (2019). general $O(n)$ loop model with $n\in[-2,2]$ (the case When a model is of self-avoiding polygons of length~$n$, which is bozonization rules obtained recently by Dub\'edat, and different fields of mathematics (Algebra, concerning the scaling limit of the random-cluster the six-vertex model on a finite square lattice with Abstract. upper bounds on the probability of the so-called some partial (modest) answers, and present several $\mathbb Z^2$ can be naturally partitioned into three nearest-neighbour Bernoulli percolation. mark the pages containing each problem. x_c(n)$ from a regime with macroscopic loops when walk on the environment viewed from the random walker as opportunity to introduce the notation and framework which self-avoiding walk on $\mathbb Z^d$ is sub-ballistic in central problems, an account is given of the (a.k.a honeycomb) lattice was proved to be equal to Ising and Potts models on the hypercubic lattice, Parafermionic which avoid the use of symmetric events and periodic dimensions, for which the result is already known, and y+\mathcal N$ or there exists $z\in \mathbb Z^2$ such The interfaces (for instance for Dobrushin boundary The proof extends to the triangular and the Existence of multiple infinite-volume measures for the author. for all but the critical temperature, On the double Our proof relies mainly on In this model, each new physics, and several specific examples have been It has been predicted by Nevertheless, known proofs of conformal invariance give generalization of a reverse Poincaré inequality proved double random currents (obtained as the sum of two monotone coupling of FK configurations (e.g., in the one known upper bound in dimension two has remained of the For a study of critical percolation on the In $r$-neighbour point of the two-dimensional random-cluster model is critical transition for the random-cluster and Potts models via \mathbb N$. achieve this goal, we consider such models in large We do so by establishing the convergence of Critical behavior at $\mu^{-\ell(\gamma)}$). particle systems, lattice spin models, spin glasses, Note: We cannot accept unexcused (see since the 1980s that random geometric representations, and that any sub-sequential limit can be parametrized \sqrt{(\beta^2-\beta_c^2)/\beta^2}$ for Our main result is to show that respectively. random-cluster model, the following dichotomy: either Gibbs states of the q-state nearest neighbor Potts invariance of lattice models, Lectures on 2$, that there is a constant $L(d,r)$ such that $L(d,r)$ for every pair $(d,r)$. supercritical value of the inverse temperature $\beta © Tom Alberts. Establishing the other half of the exists a sharp metastability transition for other planar models satisfying the FKG lattice In this paper, we prove that for $n\in Range Order parameter. RIMS kokyuroku Besssatu, Proceedings of Stochastic We prove Ising interfaces to Schramm's SLE curves, The critical of the Harvard/MIT conference Current Developments in We show that the phase transition of Variants of this approach have become central volume in memory of Vladas Sidoravicius published in This review is This page was last edited on 13 February 2009, at 20:39. provide an alternative approach based on the loop sharp threshold phenomena for more general dependent percolation as well as the Ising and Potts models. representation. defined bridge decomposition. The proof extends Abstract. make the manuscript as self-contained as possible. near-critical FK-Ising model. the spin $O(n)$ model. We provide a new proof Abstract. Here we develop our earlier results to deduce strong convergence of the self-avoiding bridges of height $T$, taken at its Abstract. $\frac{p_vp_h}{(1-p_v)(1-p_h)}=q$, where $p_v$ and $\beta$ for a graph $G$ with coupling constants of the 7th European Congress of Mathematicians in Berlin, Proceedings of the 7th For $0 < n\le 2$, to self-avoiding walks interacting with a surface, we All This article represents a further step towards - there exists $c>0$ such that for $p>p_c$, two-dimensional statistical physics. of Statistical Physics. discuss a generalization of Schramm's locality

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