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Volume 52, Number 3, 2016 ISSN 0246-0203 Volume 52, Number 3, 2016 ISSN 0246-0203 Martingale defocusing and transience of a self-interacting random walk Y. Peres, B. Schapira and P. Sousi 1009–1022 Excited random walk with periodic cookies G. Kozma, T. Orenshtein and I. Shinkar 1023–1049 Harmonic measure in the presence of a spectral gap ....I. Benjamini and A. Yadin 1050–1060 How vertex reinforced jump process arises naturally ......................X. Zeng 1061–1075 Persistence of some additive functionals of Sinai’s walk ...............A. Devulder 1076–1105 Random directed forest and the Brownian web ......R. Roy, K. Saha and A. Sarkar 1106–1143 Slowdown in branching Brownian motion with inhomogeneous variance ..............................................P. Maillard and O. Zeitouni 1144–1160 Maximal displacement of critical branching symmetric stable processes ..................................................S. P. Lalley and Y. Shao 1161–1177 On the asymptotic behavior of the density of the supremum of Lévy processes ............................................L. Chaumont and J. Małecki 1178–1195 Large deviations for non-Markovian diffusions and a path-dependent Eikonal equation ......................................J.Ma,Z.Ren,N.TouziandJ.Zhang 1196–1216 Inviscid limits for a stochastically forced shell model of turbulent flow .....................................S. Friedlander, N. Glatt-Holtz and V. Vicol 1217–1247 Estimate for Pt D for the stochastic Burgers equation G. Da Prato and A. Debussche 1248–1258 Skorokhod embeddings via stochastic flows on the space of Gaussian measures ................................................................R. Eldan 1259–1280 Liouville heat kernel: Regularity and bounds P. Maillard, R. Rhodes, V. Vargas and O. Zeitouni 1281–1320 Total length of the genealogical tree for quadratic stationary continuous-state branching processes .......................................H. Bi and J.-F. Delmas 1321–1350 Weak shape theorem in first passage percolation with infinite passage times ........................................................R. Cerf and M. Théret 1351–1381 Critical Ising model and spanning trees partition functions ...........B. de Tilière 1382–1405 An SLE2 loop measure .....................................S. Benoist and J. Dubédat 1406–1436 Independences and partial R-transforms in bi-free probability ......P. S k o u f r a n i s 1437–1473 Precise large deviation results for products of random matrices D. Buraczewski and S. Mentemeier 1474–1513 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 V. BALADI (Ecole Normale Supérieure, Paris) G. BLANCHARD (Weierstrass Inst., Berlin) T. BODINEAU (École Polytechnique) P. B OURGADE (New York Univ.) P. C APUTO (Università Roma Tre) B. COLLINS (Université d’Ottawa) I. CORWIN (Columbia University) F. DELARUE (Université de Nice Sophia-Antipolis) H. DUMINIL-COPIN (Université de Genève) F. FLANDOLI (Univ. of Pisa) G. GIACOMIN (Université Paris Diderot) M. HAIRER (Warwick Univ.) M. HOFFMANN (Univ. Paris-Dauphine) Y. H U (Université Paris 13) P. M ATHIEU (Univ. de Provence) L. MYTNIK (Israel Inst. of Technology) A. NACHMIAS (Tel Aviv University) E. PERKINS (Univ. British Columbia) G. PETE (Technical Univ. of Budapest) V. WACHTEL (Universität München) L. ZAMBOTTI (Univ. Pierre et Marie Curie) Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques (ISSN 0246-0203), Volume 52, Number 3, August 2016. 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 © 2016 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 2016, Vol. 52, No. 3, 1009–1022 DOI: 10.1214/14-AIHP667 © Association des Publications de l’Institut Henri Poincaré, 2016 Martingale defocusing and transience of a self-interacting random walk Yuval Peresa, Bruno Schapirab and Perla Sousic aMicrosoft Research, Redmond, Washington, USA. E-mail: [email protected] bAix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France. E-mail: [email protected] cUniversity of Cambridge, Cambridge, UK. E-mail: [email protected] Abstract. Suppose that (X,Y,Z) is a random walk in Z3 that moves in the following way: on the first visit to a vertex only Z changes by ±1 equally likely, while on later visits to the same vertex (X, Y ) performs a two-dimensional random walk step. We show that this walk is transient thus answering a question of Benjamini, Kozma and Schapira. One important ingredient of the proof is a dispersion result for martingales. Résumé. Supposons que (X,Y,Z)soit une marche aléatoire dans Z3 qui se déplace de la façon suivante : à la première visite en un site, seule la coordonnée Z saute de ±1 avec probabilité uniforme, et aux visites suivantes en ce site (X, Y ) effectue un saut dans l’ensemble {(±1, 0), (0, ±1)} avec probabilité uniforme. Nous montrons que cette marche est transiente, répondant ainsi à une question de Benjamini, Kozma et Schapira. Un ingrédient important de la preuve est un résultat de dispersion pour les martingales. MSC: 60K35 Keywords: Transience; Martingale; Self-interacting random walk; Excited random walk References [1] K. S. Alexander. Controlled random walk with a target site. Electron. Commun. Probab. 18 (2013) 43. MR3070909 [2] O. Angel, N. Crawford and G. Kozma. Localization for linearly edge reinforced random walks. Duke Math. J. 163 (5) (2014) 889–921. MR3189433 [3] S. N. Armstrong and O. Zeitouni. Local asymptotics for controlled martingales, 2014. Available at arXiv:1402.2402. [4] A.-L. Basdevant, B. Schapira and A. Singh. Localization of a vertex reinforced random walk on Z with sub-linear weight. Probab. Theory Related Fields 159 (1–2) (2014) 75–115. MR3201918 [5] R. F. Bass, X. Chen and J. Rosen. Moderate deviations for the range of planar random walks. Mem. Amer. Math. Soc. 198 (929) (2009) 1–82. MR2493313 [6] R. F. Bass and T. Kumagai. Laws of the iterated logarithm for the range of random walks in two and three dimensions. Ann. Probab. 30 (3) (2002) 1369–1396. MR1920111 [7] I. Benjamini, G. Kozma and B. Schapira. A balanced excited random walk. C. R. Math. Acad. Sci. Paris 349 (7–8) (2011) 459–462. MR2788390 [8] I. Benjamini and D. B. Wilson. Excited random walk. Electron. Commun. Probab. 8 (2003) 86–92 (electronic). MR1987097 [9] R. Durrett. Probability: Theory and Examples, 4th edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge Univ. Press, Cambridge, 2010. MR2722836 [10] A. Erschler, B. Tóth and W. Werner. Stuck walks. Probab. Theory Related Fields 154 (1–2) (2012) 149–163. MR2981420 [11] O. Gurel-Gurevich, Y. Peres and O. Zeitouni. Localization for controlled random walks and martingales. Electron. Commun. Probab. 19 (2014) 24. MR3208322 [12] E. Kosygina and M. P. W. Zerner. Excited random walks: Results, methods, open problems. Bull. Inst. Math. Acad. Sin. (N.S.) 8 (1) (2013) 105–157. MR3097419 [13] G. F. Lawler and V. Limic. Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics 123. Cambridge Univ. Press, Cambridge, 2010. MR2677157 [14] D. A. Levin, Y. Peres and E. L. Wilmer. Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI, 2009. With a chapter by Propp, James G. and Wilson, David B. MR2466937 [15] F. Merkl and S. W. W. Rolles. Recurrence of edge-reinforced random walk on a two-dimensional graph. Ann. Probab. 37 (5) (2009) 1679– 1714. MR2561431 [16] R. Pemantle. A survey of random processes with reinforcement. Probab. Surv. 4 (2007) 1–79. MR2282181 [17] Y. Peres, S. Popov and P. Sousi. On recurrence and transience of self-interacting random walks. Bull. Braz. Math. Soc. (N.S.) 44 (4) (2013) 841–867. MR3167134 [18] O. Raimond and B. Schapira. Random walks with occasionally modified transition probabilities. Illinois J. Math. 54 (4) (2012) 1213–1238. 2010. MR2981846 [19] C. Sabot and P. Tarres. Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. J. Eur. Math. Soc. 17 (2015) 2353–2378. MR3420510 [20] P. Tarrès. Vertex-reinforced random walk on Z eventually gets stuck on five points. Ann. Probab. 32 (3B) (2004) 2650–2701. MR2078554 [21] B. Tóth. Self-interacting random motions. In European Congress of Mathematics I 555–564. Barcelona, 2000. Progr. Math. 201. Birkhäuser, Basel, 2001. MR1905343 Annales de l’Institut Henri Poincaré - Probabilités et Statistiques 2016, Vol. 52, No. 3, 1023–1049 DOI: 10.1214/15-AIHP669 © Association des Publications de l’Institut Henri Poincaré, 2016 Excited random walk with periodic cookies Gady Kozma1,3, Tal Orenshtein1 and Igor Shinkar2 Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: [email protected]; [email protected]; [email protected] Abstract. In this paper we consider an excited random walk (ERW) on Z in identically piled periodic environment. This is a M discrete time process on Z defined by parameters (p1,...,pM ) ∈[0, 1] for some positive integer M, where the walker upon the ith visit to z ∈ Z moves to z + 1 with probability pi(mod M), and moves to z − 1 with probability 1 − pi(mod M).Wegivean explicit formula in terms of the parameters (p1,...,pM ) which determines whether the walk is recurrent, transient to the left, or 1 M = 1 transient to the right. In particular, in the case that M i=1 pi 2 all behaviors are possible, and may depend on the order of the pi. Our framework allows us to reprove some known results on ERW and branching processes with migration with no additional effort.
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