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Volume 56, Number 4, 2020 ISSN 0246-0203 Volume 56, Number 4, 2020 ISSN 0246-0203 Self-avoiding walk on Z2 with Yang–Baxter weights: Universality of critical fugacity and 2-point function A. Glazman and I. Manolescu 2281–2300 Hausdorff dimension of the uniform measure of Galton–Watson trees without the XlogX condition E. Aïdékon 2301–2306 Sparse random matrices have simple spectrum ...............................................K. Luh and V. Vu 2307–2328 Hard-edge asymptotics of the Jacobi growth process ..................................M. Cerenzia and J. Kuan 2329–2355 Hanson–Wright inequality in Banach spaces ..............................R. Adamczak, R. Latała and R. Meller 2356–2376 Sparse space–time models: Concentration inequalities and Lasso ................G. Ost and P. Reynaud-Bouret 2377–2405 Fluctuation lower bounds in planar random growth models .........................E. Bates and S. Chatterjee 2406–2427 Entropy and expansion........................................................E. Csóka, V. Harangi and B. Virág 2428–2444 Fluctuations of Biggins’ martingales at complex parameters ............A.Iksanov,K.KoleskoandM.Meiners 2445–2479 A natural extension of Markov processes and applications to singular SDEs L.Beznea,I.CîmpeanandM.Röckner 2480–2506 Lower deviation and moderate deviation probabilities for maximum of a branching random walk X. Chen and H. He 2507–2539 Functional approximations via Stein’s method of exchangeable pairs ..........................M. J. Kasprzak 2540–2564 Subordination methods for free deconvolution .........................O. Arizmendi, P. Tarrago and C. Vargas 2565–2594 Long-time limits and occupation times for stable Fleming–Viot processes with decaying sampling rates M. A. Kouritzin and K. Lê 2595–2620 Cutoff for the Bernoulli–Laplace urn model with o(n) swaps .....................A. Eskenazis and E. Nestoridi 2621–2639 Semi-Markov processes, integro-differential equations and anomalous diffusion-aggregation M. Savov and B. Toaldo 2640–2671 Comparing with octopi ...................................................................G. Alon and G. Kozma 2672–2685 On the convergence of random tridiagonal matrices to stochastic semigroups .......P. Y. Gaudreau Lamarre 2686–2731 Induced graphs of uniform spanning forests ......................................R. Lyons, Y. Peres and X. Sun 2732–2744 Cutoff for random walk on dynamical Erdos–Rényi˝ graph ..............................P. Sousi and S. Thomas 2745–2773 Convergence of local supermartingales ...................................................M. Larsson and J. Ruf 2774–2791 On the Poisson boundary of the relativistic Brownian motion ...........................J. Angst and C. Tardif 2792–2821 Eigenvectors distribution and quantum unique ergodicity for deformed Wigner matrices .........L. Benigni 2822–2867 Obliquely reflected backward stochastic differential equations ................J.-F. Chassagneux and A. Richou 2868–2896 Free energy of bipartite spherical Sherrington–Kirkpatrick model ........................J. Baik and J. O. Lee 2897–2934 The random transposition dynamics on random regular graphs and the Gaussian free field S. Ganguly and S. Pal 2935–2970 Spectral gap of sparse bistochastic matrices with exchangeable rows ......C. Bordenave, Y. Qiu and Y. Zhang 2971–2995 A functional limit theorem for coin tossing Markov chains .............S. Ankirchner, T. Kruse and M. Urusov 2996–3019 A Central Limit Theorem for the stochastic wave equation with fractional noise F.Delgado-Vences,D.NualartandG.Zheng 3020–3042 56 4 ANNALES DE L’INSTITUT HENRI POINCARÉ PROBABILITÉS ET STATISTIQUES Vol. 56, No. 4 (November, 2020) 2281–3042 Rédacteurs en chef / Chief Editors Grégory MIERMONT École Normale Supérieure de Lyon CNRS UMR 5669, Unité de Mathématiques Pures et Appliquées 46, allée d’Italie 69364 Lyon Cedex 07, France [email protected] Christophe SABOT Université Claude Bernard Lyon 1 CNRS UMR 5208, Institut Camille Jordan 43 blvd. du 11 novembre 1918 69622 Villeurbanne cedex, France [email protected] Comité de Rédaction / Editorial Board S. ARLOT (Université Paris-Sud) G. BLANCHARD (Weierstrass Inst., Berlin) T. BODINEAU (École Polytechnique) P. B OURGADE (New York Univ.) P. C APUTO (Università Roma Tre) F. CARAVENNA (Univ. Milano-Bicocca) B. COLLINS (Kyoto University) I. CORWIN (Columbia University) F. DELARUE (Université de Nice Sophia-Antipolis) H. DUMINIL-COPIN (Institut des Hautes Études Scientifiques) F. FLANDOLI (Univ. of Pisa) B. GESS (Universität Bielefeld) S. GOUËZEL (Université de Nantes) M. HAIRER (Imperial College London) M. HOFFMANN (Univ. Paris-Dauphine) M. HOFMANOVÁ (Bielefeld University) Y. H U (Université Paris 13) P. M ATHIEU (Univ. de Provence) A. NACHMIAS (Tel Aviv University) J. NORRIS (Cambridge University) G. PETE (Technical Univ. of Budapest) B. DE TILIÈRE (Univ. Paris-Dauphine) F. TONINELLI (CNRS, Université Claude Bernard Lyon 1) V. WACHTEL (Universität München) H. WEBER (Univ. of Warwick) L. ZAMBOTTI (Sorbonne Université (LPSM)) Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques (ISSN 0246-0203), Volume 56, Number 4, November 2020. Published quarterly by Association des Publications de l’Institut Henri Poincaré. POSTMASTER: Send address changes to Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques, Dues and Subscriptions Office, 9650 Rockville Pike, Suite L 2310, Bethesda, Maryland 20814-3998 USA. Copyright © 2020 Association des Publications de l’Institut Henri Poincaré Président et directeur de la publication : Cédric Villani Printed in the United States of America Périodicité : 4 nos /an Annales de l’Institut Henri Poincaré - Probabilités et Statistiques 2020, Vol. 56, No. 4, 2281–2300 https://doi.org/10.1214/19-AIHP1024 © Association des Publications de l’Institut Henri Poincaré, 2020 Self-avoiding walk on Z2 with Yang–Baxter weights: Universality of critical fugacity and 2-point function Alexander Glazmana and Ioan Manolescub aSchool of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel. E-mail: [email protected] bUniversité de Fribourg, Fribourg, Switzerland. E-mail: [email protected] Abstract. We consider a self-avoiding walk model (SAW) on the faces of the square lattice Z2. 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 ∈[π 2π ] weight that depends on the way the walk passes through it. The local weights are parametrised by angles θ 3 , 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 concides with that of the self-avoiding walk on the hexagonal π lattice. Indeed, the latter can be obtained by choosing all angles θ equal to 3 . √ For the hexagonal lattice, the critical fugacity of SAW was recently proved to be equal to 1 + 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. Résumé. On considère un modèle de marches auto-évitantes sur les faces du réseau carré Z2. Ce type de marche peut traverser la même face deux fois, mais traverse chaque arrête au plus une fois. Le poids d’une telle marche est le produit de poids locaux : chaque face visitée contribue par un poids qui dépend de la façon dont la marche la traverse. Les poids locaux associés à chaque face sont ∈[π 2π ] paramétrés par des angles θ 3 , 3 et satisfont l’équation de Yang–Baxter. La marche est plongée dans le plan en remplaçant les faces carrées du réseau par des losanges d’angles correspondant à leur poids. À l’aide de la transformation de Yang–Baxter, on montre que la fonction à deux points de la marche dans le demi-plan ne dépend pas des angles des losanges. En particulier, cette statistique coïncide avec celle de la marche aléatoire sur le réseau hexagonal – celle-ci π est obtenue en choisissant tous les angles θ égaux à 3 . √ La fugacité critique des marches auto-évitantes sur le réseau hexagonal a été calculée récemment : elle vaut 1 + 2. Nous montrons que la même chose est valable pour tout choix d’angles. A cette occasion, on donne une nouvelle preuve du fait que la fonction de partition des ponts auto-évitants dans une bande du réseau hexagonal tend vers 0 quand la largeur de la bande tend vers l’infini. De plus, on montre une borne quantitative sur le taux de convergence. MSC2020 subject classifications: 60K35; 60D05; 82B23; 82B20 Keywords: Self-avoiding walk; Yang–Baxter; Universality; Isoradial graphs; Rhombic tiling; Critical fugacity References [1] I. T. Alam and M. T. Batchelor. Integrability as a consequence of discrete holomorphicity: Loop models. J. Phys. A 47 (21) (2014) 215201. MR3206123 https://doi.org/10.1088/1751-8113/47/21/215201 [2] N. R. Beaton, M. Bousquet-Mélou, J. de√ Gier, H. Duminil-Copin and A. J. Guttmann. The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is 1 + 2. Comm. Math. Phys. 326 (3) (2014) 727–754. MR3173404 https://doi.org/10.1007/s00220-014-1896-1 [3] D. Chelkak and S. Smirnov. Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189 (3) (2012) 515–580. MR2957303 https://doi.org/10.1007/s00222-011-0371-2 [4] H. Duminil-Copin, J.-H. Li and I. Manolescu. Universality for the random-cluster model on isoradial graphs. Electron. J. Probab. 23 (2018) 96. [5] H. Duminil-Copin and S. Smirnov. Conformal invariance of lattice models. In Probability and Statistical Physics in Two and More Dimensions 213–276. Clay Math. Proc. 15. Amer. Math. Soc., Providence, RI, 2012. MR3025392 √ [6] H. Duminil-Copin and S. Smirnov. The connective constant of the honeycomb lattice equals 2 + 2. Ann. of Math. 175 (3) (2012) 1653–1665. MR2912714 https://doi.org/10.4007/annals.2012.175.3.14 [7] A.
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