Journal: Annales scientifiques de l'École Normale Supérieure

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Société Mathématique de France

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0012-9593
1873-2151

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2019 - 201912020 - 20233
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Publications 1 - 4 of 4
  • Discontinuity of the Phase Transition for the Planar Random-Cluster and Potts Models with q > 4
    Item type: Journal Article
    Duminil-Copin, Hugo; Gagnebin, Maxime; Harel, Matan; et al. (2021)
    Annales scientifiques de l'École Normale Supérieure
    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(π2/√(q−4)) as q tends to 4.
  • Hochschild-Pirashvili homology on suspensions and representations of Out(Fn)
    Item type: Journal Article
    Turchin, Victor; Willwacher, Thomas (2019)
    Annales scientifiques de l'École Normale Supérieure
  • On Bulk Deviations for the Local Behavior of Random Interlacements
    Item type: Journal Article
    Sznitman, Alain-Sol (2023)
    Annales scientifiques de l'École Normale Supérieure
    We investigate certain large deviation asymptotics concerning random interlacements in ℤᵈ, d ≥ 3. We find the principal exponential rate of decay for the probability that the average value of some suitable non-decreasing local function of the field of occupation times, sampled at each point of a large box, exceeds its expected value. We express the exponential rate of decay in terms of a constrained minimum for the Dirichlet energy of functions on ℝᵈ that decay at infinity. An application concerns the excess presence of random interlacements in a large box. Our findings exhibit similarities to some of the results of van den Berg-Bolthausen-den Hollander in their work on moderate deviations of the volume of the Wiener sausage. An other application relates to recent work of the author on macroscopic holes in connected components of the vacant set.
  • A sharp Freiman type estimate for semisums in two and three dimensional Euclidean spaces
    Item type: Journal Article
    Figalli, Alessio; Jerison, David (2021)
    Annales scientifiques de l'École Normale Supérieure
    Freiman's theorem is a classical result in additive combinatorics concerning the approximate structure of sets of integers that contain a high proportion of their internal sums. As a consequence, one can deduce an estimate for sets of real numbers: "If A⊂R and ∣∣12(A+A)∣∣−|A|≪|A|, then A is close to its convex hull.'' In this paper we prove a sharp form of the analogous result in dimensions 2 and 3. © 2021 Société Mathématique de France
Publications 1 - 4 of 4