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Volume 57, Number 3, 2021 ISSN 0246-0203 Volume 57, Number 3, 2021 ISSN 0246-0203 The mean-field quantum Heisenberg ferromagnet via representation theory .......G. Alon and G. Kozma 1203–1228 Conformal welding for critical Liouville quantum gravity...........................N. Holden and E. Powell 1229–1254 U-bootstrap percolation: Critical probability, exponential decay and applications ............I. Hartarsky 1255–1280 The TAZRP speed process ...................................G. Amir, O. Busani, P. Gonçalves and J. B. Martin 1281–1305 Cutoff for polymer pinning dynamics in the repulsive phase .......................................S. Yang 1306–1335 On a stochastic representation theorem for Meyer-measurable processes ..........P. Bank and D. Besslich 1336–1368 Non-uniqueness for reflected rough differential equations........................................P. Gassiat 1369–1387 Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds D.Barilari,U.Boscain,D.CannarsaandK.Habermann 1388–1410 Persistence exponents in Markov chains ...........................F. Aurzada, S. Mukherjee and O. Zeitouni 1411–1441 Piecewise deterministic Markov processes and their invariant measures A. Durmus, A. Guillin and P. Monmarché 1442–1475 On consistent and rate optimal estimation of the missing mass F.Ayed,M.Battiston,F.CamerlenghiandS.Favaro 1476–1494 Adaptive regression with Brownian path covariate ............................K. Bertin and N. Klutchnikoff 1495–1520 Spike and slab Pólya tree posterior densities: Adaptive inference..................I. Castillo and R. Mismer 1521–1548 Strongly vertex-reinforced jump process on a complete graph ...............O. Raimond and T.-M. Nguyen 1549–1568 Uniform spanning forests on biased Euclidean lattices Z. Shi, V. Sidoravicius, H. Song, L. Wang and K. Xiang 1569–1582 Testing degree corrections in stochastic block models..............................R. Mukherjee and S. Sen 1583–1635 Incompressible viscous fluids in R2 and SPDEs on graphs, in presence of fast advection and non smooth noise..........................................................................S. Cerrai and G. Xi 1636–1664 Exponential ergodicity for SDEs and McKean–Vlasov processes with Lévy noise M. Liang, M. B. Majka and J. Wang 1665–1701 Asymptotics of PDE in random environment by paracontrolled calculus T.Funaki,M.Hoshino,S.SethuramanandB.Xie 1702–1735 Probabilistic potential theory and induction of dynamical systems ................F. Pène and D. Thomine 1736–1767 Longest common substring for random subshifts of finite type .................................J. Rousseau 1768–1785 Derivative martingale of the branching Brownian motion in dimension d ≥ 1 R. Stasi´nski,J.BerestyckiandB.Mallein 1786–1810 57 3 ANNALES DE L’INSTITUT HENRI POINCARÉ PROBABILITÉS ET STATISTIQUES Vol. 57, No. 3 (August, 2021) 1203–1810 Rédacteurs en chef / Chief Editors Giambattista GIACOMIN Université de Paris 5 rue Thomas Mann, 75013 Paris, France [email protected] Yueyun HU Université Sorbonne Paris Nord 99 Av. J-B Clément, 93430 Villetaneuse, France [email protected] Comité de Rédaction / Editorial Board E. AIDEKON (Fudan University) S. ARLOT (Université Paris-Sud) J. BERTOIN (Universität Zürich) G. BLANCHARD (Université Paris-Sud) P. B OURGADE (New York Univ.) F. CARAVENNA (Univ. Milano-Bicocca) B. COLLINS (Kyoto University) I. CORWIN (Columbia University) A. DEBUSSCHE (École Normale Supérieure de Rennes) B. GESS (Universität Bielefeld) S. GOUËZEL (Université de Nantes) A. GUILLIN (Clermont-Auvergne University) M. HAIRER (Imperial College London) M. HOFFMANN (Univ. Paris-Dauphine) N. HOLDEN (ETH Zurich) T. HUTCHCROFT (Cambridge University) P. M ATHIEU (Univ. de Provence) A. NACHMIAS (Tel Aviv University) J. NORRIS (Cambridge University) G. PETE (Technical Univ. of Budapest) R. RHODES (Univ. Aix-Marseille) M. SASADA (University of Tokyo) P. S OUSI (Cambridge University) B. DE TILIÈRE (Univ. Paris-Dauphine) V. WACHTEL (Universität München) H. WEBER (Univ. of Warwick) Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques (ISSN 0246-0203), Volume 57, Number 3, August 2021. 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, PO Box 729, Middletown, Maryland 21769, USA. Copyright © 2021 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 2021, Vol. 57, No. 3, 1203–1228 https://doi.org/10.1214/20-AIHP1067 © Association des Publications de l’Institut Henri Poincaré, 2021 The mean-field quantum Heisenberg ferromagnet via representation theory Gil Alona and Gady Kozmab aDepartment of Mathematics and Computer Science, The Open University of Israel, 4353701 Raanana, Israel. E-mail: [email protected] bDepartment of Mathematics and Computer Science, The Weizmann Institute of Science, 76100 Rehovot, Israel. E-mail: [email protected] Abstract. We use representation theory to write a formula for the magnetisation of the quantum Heisenberg ferromagnet. The core α +···+α new result is a spectral decomposition of the function αk2 1 n where αk is the number of cycles of length k of a permutation. In the mean-field case, we simplify the formula further, arriving at a closed-form expression for the magnetisation, which allows to analyse the phase transition. Résumé. À l’aide de la théorie des représentations, nous donnons une formule pour la magnétisation du ferro-aimant de Heisenberg α +···+α quantique. La nouveauté-clé est une décomposition spectrale de la fonction αk2 1 n où αk est le nombre de cycles de longueur k d’une permutation. Dans le cas à champ moyen, nous simplifions encore la formule, ce qui donne une expression fermée pour la magnétisation qui permet d’analyser la transition de phase. MSC2020 subject classifications: 82C22; 60B15; 20C30 Keywords: Quantum heizenberg ferromagnet; Interchange process; Random walk; Symmetric group; Magnetisation; Phase transition References [1] R. Adamczak, M. Kotowski and P. Miłos.´ Phase transition for the interchange and quantum Heisenberg models on the Hamming graph. Ann. Inst. Henri Poincaré Probab. Stat. 57 (1) (2021) 273–325. MR4255176 https://doi.org/10.1214/20-aihp1079 [2] G. Alon and G. Kozma. The probability of long cycles in interchange processes. Duke Math. J. 162 (9) (2013) 1567–1585. [3] G. Alon and G. Kozma. Comparing with octopi. Ann. Inst. Henri Poincaré. To appear. [4] O. Angel. Random infinite permutations and the cyclic time random walk Discrete random walks (Paris, 2003), 9–16, Discrete Math. Theor. Comput. Sci. Proc. Conference Vol. AC. Available at emis.ams/dmAC0101. [5] N. Berestycki and R. Durrett. A phase transition in the random transposition random walk. Probab. Theory Related Fields 136 (2) (2006) 203–233. MR2240787 https://doi.org/10.1007/s00440-005-0479-7 [6] N. Berestycki and G. Kozma. Cycle structure of the interchange process and representation theory. Bull. Soc. Math. France 143 (2) (2015) 265–280. MR3351179 https://doi.org/10.24033/bsmf.2686 [7] J. E. Björnberg. Large cycles in random permutation related to the Heisenberg model. Electron. Commun. Probab. 20 (2015) 55. [8] J. E. Björnberg. The free energy in a class of quantum spin systems and interchange processes. J. Math. Phys. 57 (2016) 073303. [9] J. G. Conlon and J. P. Solovej. Random walk representations of the Heisenberg model. J. Stat. Phys. 64 (1–2) (1991) 251–270. [10] M. Correggi, A. Giuliani and R. Seiringer. Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet. Comm. Math. Phys. 339 (1) (2015) 279–307. MR3366059 https://doi.org/10.1007/s00220-015-2402-0 [11] P. Diaconis. Applications of noncommutative Fourier analysis to probability problems. In École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87 51–100. Lecture Notes in Math. 1362. Springer, Berlin, 1988. MR0983372 https://doi.org/10.1007/BFb0086179 [12] P. Diaconis and L. Saloff-Coste. Comparison techniques for random walk on finite groups. Ann. Probab. 21 (4) (1993) 2131–2156. [13] P. Diaconis and M. Shahshahani. Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57 (2) (1981) 159–179. MR0626813 https://doi.org/10.1007/BF00535487 [14] F. J. Dyson, E. H. Lieb and B. Simon. Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. J. Stat. Phys. 18 (4) (1978) 335–383. MR0496246 https://doi.org/10.1007/BF01106729 [15] G. F. Frobenius. Über die Charaktere der symmetrischen Gruppe. [German, On the characters of the symmetric group]. In Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin 516–534, 1900. [16] J. Fröhlich, R. Israel, E. H. Lieb and B. Simon. Phase transitions and reflection positivity. I. General theory and long range lattice models. Comm. Math. Phys. 62 (1) (1978) 1–34. [17] J. Fröhlich, R. Israel, E. H. Lieb and B. Simon. Phase transitions and reflection positivity. II. Lattice systems with short-range and Coulomb interactions. J. Stat. Phys. 22 (3) (1980) 297–347. MR0570370 https://doi.org/10.1007/BF01014646 [18] J. Fröhlich, B. Simon and T. Spencer. Infrared bounds, phase transitions and continuous symmetry breaking. Comm. Math. Phys. 50 (1) (1976) 79–95. [19] J. Gordon and A. Kerber. The Representation Theory of the Symmetric Group. Encyclopedia
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