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Localization of directed polymers with general reference walk. (English) Zbl 1390.60359

Summary: Directed polymers in random environment have usually been constructed with a simple random walk on the integer lattice. It has been observed before that several standard results for this model continue to hold for a more general reference walk. Some finer results are known for the so-called long-range directed polymer in which the reference walk lies in the domain of attraction of an \(\alpha\)-stable process. In this note, low-temperature localization properties recently proved for the classical case are shown to be true with any reference walk. First, it is proved that the polymer’s endpoint distribution is asymptotically purely atomic, thus strengthening the best known result for long-range directed polymers. A second result proving geometric localization along a positive density subsequence is new to the general case. The proofs use a generalization of the approach introduced by the author with S. Chatterjee in a recent manuscript on the quenched endpoint distribution; this generalization allows one to weaken assumptions on the both the walk and the environment. The methods of this paper also give rise to a variational formula for free energy which is analogous to the one obtained in the simple random walk case.

MSC:

60K37 Processes in random environments
82B26 Phase transitions (general) in equilibrium statistical mechanics
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82D60 Statistical mechanics of polymers
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[1] Tom Alberts, Konstantin Khanin, and Jeremy Quastel, The continuum directed random polymer, J. Stat. Phys. 154 (2014), no. 1-2, 305-326. · Zbl 1291.82143 · doi:10.1007/s10955-013-0872-z
[2] Tom Alberts, Konstantin Khanin, and Jeremy Quastel, The intermediate disorder regime for directed polymers in dimension \(1+1\), Ann. Probab. 42 (2014), no. 3, 1212-1256. · Zbl 1292.82014 · doi:10.1214/13-AOP858
[3] Kenneth S. Alexander and Gökhan Yıldırım, Directed polymers in a random environment with a defect line, Electron. J. Probab. 20 (2015), no. 6, 20. · Zbl 1308.82038
[4] Antonio Auffinger and Wei-Kuo Chen, On properties of Parisi measures, Probab. Theory Related Fields 161 (2015), no. 3-4, 817-850. · Zbl 1322.60204 · doi:10.1007/s00440-014-0563-y
[5] Antonio Auffinger and Wei-Kuo Chen, The Parisi formula has a unique minimizer, Comm. Math. Phys. 335 (2015), no. 3, 1429-1444. · Zbl 1320.82033 · doi:10.1007/s00220-014-2254-z
[6] Antonio Auffinger and Wei-Kuo Chen, The Legendre structure of the Parisi formula, Comm. Math. Phys. 348 (2016), no. 3, 751-770. · Zbl 1362.82029 · doi:10.1007/s00220-016-2673-0
[7] Antonio Auffinger and Wei-Kuo Chen, Parisi formula for the ground state energy in the mixed \(p\)-spin model, Ann. Probab. 45 (2017), no. 6B, 4617-4631. · Zbl 1393.60116 · doi:10.1214/16-AOP1173
[8] Ole E Barndorff-Nielsen, Thomas Mikosch, and Sidney I Resnick, Lévy Processes: Theory and Applications, Birkhäuser, Boston, 2001. · Zbl 0961.00012
[9] Julien Barral, Rémi Rhodes, and Vincent Vargas, Limiting laws of supercritical branching random walks, C. R. Math. Acad. Sci. Paris 350 (2012), no. 9-10, 535-538. · Zbl 1260.60173 · doi:10.1016/j.crma.2012.05.013
[10] Erik Bates and Sourav Chatterjee, The endpoint distribution of directed polymers, Preprint, available at arXiv:1612.03443. · Zbl 1444.60087
[11] Quentin Berger and Hubert Lacoin, The high-temperature behavior for the directed polymer in dimension \(1+2\), Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017), no. 1, 430-450. · Zbl 1362.82055
[12] Patrick Billingsley, Convergence of probability measures, second ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999, A Wiley-Interscience Publication. · Zbl 0944.60003
[13] Stéphane Boucheron, Gábor Lugosi, and Pascal Massart, Concentration inequalities, Oxford University Press, Oxford, 2013, A nonasymptotic theory of independence, With a foreword by Michel Ledoux. · Zbl 1337.60003
[14] D. Brockmann and T. Geisel, Particle dispersion on rapidly folding random heteropolymers, Phys. Rev. Lett. 91 (2003), 048303-048306.
[15] D. L. Burkholder, B. J. Davis, and R. F. Gundy, Integral inequalities for convex functions of operators on martingales, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, Univ. California Press, Berkeley, Calif., 1972, pp. 223-240. · Zbl 0253.60056
[16] Francesco Caravenna, Rongfeng Sun, and Nikos Zygouras, Polynomial chaos and scaling limits of disordered systems, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 1, 1-65. · Zbl 1364.82026 · doi:10.4171/JEMS/660
[17] Francesco Caravenna, Rongfeng Sun, and Nikos Zygouras, Universality in marginally relevant disordered systems, Ann. Appl. Probab. 27 (2017), no. 5, 3050-3112. · Zbl 1387.82032 · doi:10.1214/17-AAP1276
[18] Philippe Carmona and Yueyun Hu, On the partition function of a directed polymer in a Gaussian random environment, Probab. Theory Related Fields 124 (2002), no. 3, 431-457. · Zbl 1015.60100 · doi:10.1007/s004400200213
[19] Ligang Chen and Michael W. Deem, Reaction, lévy flights, and quenched disorder, Phys. Rev. E 65 (2001), 011109-011114.
[20] Wei-Kuo Chen, The Aizenman-Sims-Starr scheme and Parisi formula for mixed \(p\)-spin spherical models, Electron. J. Probab. 18 (2013), no. 94, 14. · Zbl 1288.60127
[21] Wei-Kuo Chen, Variational representations for the Parisi functional and the two-dimensional Guerra-Talagrand bound, Ann. Probab. 45 (2017), no. 6A, 3929-3966. · Zbl 1409.60139 · doi:10.1214/16-AOP1154
[22] F. Comets, Weak disorder for low dimensional polymers: the model of stable laws, Markov Process. Related Fields 13 (2007), no. 4, 681-696. · Zbl 1144.60059
[23] Francis Comets, Directed polymers in random environments, Lecture Notes in Mathematics, vol. 2175, Springer, Cham, 2017, Lecture notes from the 46th Probability Summer School held in Saint-Flour, 2016. · Zbl 1392.60002
[24] Francis Comets, Ryoki Fukushima, Shuta Nakajima, and Nobuo Yoshida, Limiting results for the free energy of directed polymers in random environment with unbounded jumps, J. Stat. Phys. 161 (2015), no. 3, 577-597. · Zbl 1332.82093 · doi:10.1007/s10955-015-1347-1
[25] Francis Comets and Vu-Lan Nguyen, Localization in log-gamma polymers with boundaries, Probab. Theory Related Fields 166 (2016), no. 1-2, 429-461. · Zbl 1350.60109
[26] Francis Comets, Tokuzo Shiga, and Nobuo Yoshida, Directed polymers in a random environment: path localization and strong disorder, Bernoulli 9 (2003), no. 4, 705-723. · Zbl 1042.60069 · doi:10.3150/bj/1066223275
[27] Francis Comets and Vincent Vargas, Majorizing multiplicative cascades for directed polymers in random media, ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006), 267-277. · Zbl 1105.60074
[28] Francis Comets and Nobuo Yoshida, Brownian directed polymers in random environment, Comm. Math. Phys. 254 (2005), no. 2, 257-287. · Zbl 1128.60089 · doi:10.1007/s00220-004-1203-7
[29] Francis Comets and Nobuo Yoshida, Directed polymers in random environment are diffusive at weak disorder, Ann. Probab. 34 (2006), no. 5, 1746-1770. · Zbl 1104.60061 · doi:10.1214/009117905000000828
[30] Francis Comets and Nobuo Yoshida, Localization transition for polymers in Poissonian medium, Comm. Math. Phys. 323 (2013), no. 1, 417-447. · Zbl 1276.82066 · doi:10.1007/s00220-013-1744-8
[31] Nicos Georgiou, Firas Rassoul-Agha, and Timo Seppäläinen, Variational formulas and cocycle solutions for directed polymer and percolation models, Comm. Math. Phys. 346 (2016), no. 2, 741-779. · Zbl 1355.82064 · doi:10.1007/s00220-016-2613-z
[32] Geoffrey Grimmett, Percolation, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, Springer-Verlag, Berlin, 1999. · Zbl 0926.60004
[33] David A Huse and Christopher L Henley, Pinning and roughening of domain walls in Ising systems due to random impurities, Phys. Rev. Lett. 54 (1985), no. 25, 2708-2711.
[34] J. Z. Imbrie and T. Spencer, Diffusion of directed polymers in a random environment, J. Statist. Phys. 52 (1988), no. 3-4, 609-626. · Zbl 1084.82595
[35] Aukosh Jagannath and Ian Tobasco, A dynamic programming approach to the Parisi functional, Proc. Amer. Math. Soc. 144 (2016), no. 7, 3135-3150. · Zbl 1376.60081 · doi:10.1090/proc/12968
[36] Hubert Lacoin, New bounds for the free energy of directed polymers in dimension \(1+1\) and \(1+2\), Comm. Math. Phys. 294 (2010), no. 2, 471-503. · Zbl 1227.82098 · doi:10.1007/s00220-009-0957-3
[37] Hubert Lacoin, Influence of spatial correlation for directed polymers, Ann. Probab. 39 (2011), no. 1, 139-175. · Zbl 1208.82084 · doi:10.1214/10-AOP553
[38] Paul Lévy, Théorie de L’addition des Variables Aléatoires, Gauthier-Villars, Paris, 1954. · Zbl 0016.17003
[39] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109-145. · Zbl 0541.49009
[40] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223-283. · Zbl 0704.49004
[41] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145-201. · Zbl 0704.49005 · doi:10.4171/RMI/6
[42] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoamericana 1 (1985), no. 2, 45-121. · Zbl 0704.49006 · doi:10.4171/RMI/12
[43] Quansheng Liu and Frédérique Watbled, Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment, Stochastic Process. Appl. 119 (2009), no. 10, 3101-3132. · Zbl 1177.60043 · doi:10.1016/j.spa.2009.05.001
[44] Mitsuharu Miura, Yoshihiro Tawara, and Kaneharu Tsuchida, Strong and weak disorder for Lévy directed polymers in random environment, Stoch. Anal. Appl. 26 (2008), no. 5, 1000-1012. · Zbl 1151.60350 · doi:10.1080/07362990802286418
[45] Chiranjib Mukherjee, Alexander Shamov, and Ofer Zeitouni, Weak and strong disorder for the stochastic heat equation and continuous directed polymers in \(d≥ 3\), Electron. Commun. Probab. 21 (2016), Paper No. 61, 12. · Zbl 1348.60094
[46] Chiranjib Mukherjee and S. R. S. Varadhan, Brownian occupation measures, compactness and large deviations, Ann. Probab. 44 (2016), no. 6, 3934-3964. · Zbl 1364.60037 · doi:10.1214/15-AOP1065
[47] Makoto Nakashima, A remark on the bound for the free energy of directed polymers in random environment in \(1+2\) dimension, J. Math. Phys. 55 (2014), no. 9, 093304, 14. · Zbl 1302.82135
[48] Dmitry Panchenko, A question about the Parisi functional, Electron. Comm. Probab. 10 (2005), 155-166. · Zbl 1136.82332
[49] Dmitry Panchenko, On differentiability of the Parisi formula, Electron. Commun. Probab. 13 (2008), 241-247. · Zbl 1205.82092
[50] Dmitry Panchenko, The Parisi ultrametricity conjecture, Ann. of Math. (2) 177 (2013), no. 1, 383-393. · Zbl 1270.60060 · doi:10.4007/annals.2013.177.1.8
[51] Dmitry Panchenko, The Sherrington-Kirkpatrick model, Springer Monographs in Mathematics, Springer, New York, 2013. · Zbl 1266.82005
[52] Dmitry Panchenko, The Parisi formula for mixed \(p\)-spin models, Ann. Probab. 42 (2014), no. 3, 946-958. · Zbl 1292.82020
[53] K. R. Parthasarathy, R. Ranga Rao, and S. R. S. Varadhan, On the category of indecomposable distributions on topological groups, Trans. Amer. Math. Soc. 102 (1962), 200-217. · Zbl 0104.36205
[54] Firas Rassoul-Agha, Timo Seppäläinen, and Atilla Yilmaz, Quenched free energy and large deviations for random walks in random potentials, Comm. Pure Appl. Math. 66 (2013), no. 2, 202-244. · Zbl 1267.60110 · doi:10.1002/cpa.21417
[55] Firas Rassoul-Agha, Timo Seppäläinen, and Atilla Yilmaz, Variational formulas and disorder regimes of random walks in random potentials, Bernoulli 23 (2017), no. 1, 405-431. · Zbl 1368.60105
[56] Timo Seppäläinen, Scaling for a one-dimensional directed polymer with boundary conditions, Ann. Probab. 40 (2012), no. 1, 19-73. · Zbl 1254.60098
[57] I. M. Sokolov, J. Mai, and A. Blumen, Paradoxal diffusion in chemical space for nearest-neighbor walks over polymer chains, Phys. Rev. Lett. 79 (1997), 857-860.
[58] Michel Talagrand, The Parisi formula, Ann. of Math. (2) 163 (2006), no. 1, 221-263. · Zbl 1137.82010 · doi:10.4007/annals.2006.163.221
[59] Michel Talagrand, Mean field models for spin glasses. Volume I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 54, Springer-Verlag, Berlin, 2011, Basic examples. · Zbl 1214.82002
[60] Michel Talagrand, Mean field models for spin glasses. Volume II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 55, Springer, Heidelberg, 2011, Advanced replica-symmetry and low temperature. · Zbl 1214.82002
[61] Aad W. van der Vaart and Jon A. Wellner, Weak convergence and empirical processes, Springer Series in Statistics, Springer-Verlag, New York, 1996, With applications to statistics. · Zbl 0862.60002
[62] Vincent Vargas, Strong localization and macroscopic atoms for directed polymers, Probab. Theory Related Fields 138 (2007), no. 3-4, 391-410. · Zbl 1113.60097
[63] Cédric Villani, Optimal transport. old and new., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. · Zbl 1156.53003
[64] Frédérique Watbled, Sharp asymptotics for the free energy of \(1+1\) dimensional directed polymers in an infinitely divisible environment, Electron. Commun. Probab. 17 (2012), no. 53, 9. · Zbl 1306.60154
[65] Ran Wei, Free energy of the Cauchy directed polymer model at high temperature, Preprint, available at arXiv:1706.04530. · Zbl 1407.82070
[66] Ran Wei, On the long-range directed polymer model, J. Stat. Phys. 165 (2016), no. 2, 320-350. · Zbl 1355.82070 · doi:10.1007/s10955-016-1612-y
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