Annals of Probability

The endpoint distribution of directed polymers

Erik Bates and Sourav Chatterjee

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Probabilistic models of directed polymers in random environment have received considerable attention in recent years. Much of this attention has focused on integrable models. In this paper, we introduce some new computational tools that do not require integrability. We begin by defining a new kind of abstract limit object, called “partitioned subprobability measure,” to describe the limits of endpoint distributions of directed polymers. Inspired by a recent work of Mukherjee and Varadhan on large deviations of the occupation measure of Brownian motion, we define a suitable topology on the space of partitioned subprobability measures and prove that this topology is compact. Then using a variant of the cavity method from the theory of spin glasses, we show that any limit law of a sequence of endpoint distributions must satisfy a fixed point equation on this abstract space, and that the limiting free energy of the model can be expressed as the solution of a variational problem over the set of fixed points. As a first application of the theory, we prove that in an environment with finite exponential moment, the endpoint distribution is asymptotically purely atomic if and only if the system is in the low temperature phase. The analogous result for a heavy-tailed environment was proved by Vargas in 2007. As a second application, we prove a subsequential version of the longstanding conjecture that in the low temperature phase, the endpoint distribution is asymptotically localized in a region of stochastically bounded diameter. All our results hold in arbitrary dimensions, and make no use of integrability.

Article information

Source
Ann. Probab., Volume 48, Number 2 (2020), 817-871.

Dates
Received: June 2018
Revised: February 2019
First available in Project Euclid: 22 April 2020

Permanent link to this document
https://projecteuclid.org/euclid.aop/1587542681

Digital Object Identifier
doi:10.1214/19-AOP1376

Mathematical Reviews number (MathSciNet)
MR4089496

Zentralblatt MATH identifier
07199863

Subjects
Primary: 60K37: Processes in random environments
Secondary: 82B26: Phase transitions (general) 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 82D60: Polymers

Keywords
Directed polymer free energy disordered system phase transition

Citation

Bates, Erik; Chatterjee, Sourav. The endpoint distribution of directed polymers. Ann. Probab. 48 (2020), no. 2, 817--871. doi:10.1214/19-AOP1376. https://projecteuclid.org/euclid.aop/1587542681


Export citation

References

  • [1] Aizenman, M. and Contucci, P. (1998). On the stability of the quenched state in mean-field spin-glass models. J. Stat. Phys. 92 765–783.
  • [2] Aizenman, M., Sims, R. and Starr, S. L. (2007). Mean-field spin glass models from the cavity-ROSt perspective. In Prospects in Mathematical Physics. Contemp. Math. 437 1–30. Am. Math. Soc., Providence, RI.
  • [3] Alberts, T., Khanin, K. and Quastel, J. (2014). The intermediate disorder regime for directed polymers in dimension $1+1$. Ann. Probab. 42 1212–1256.
  • [4] Albeverio, S. and Zhou, X. Y. (1996). A martingale approach to directed polymers in a random environment. J. Theoret. Probab. 9 171–189.
  • [5] Aldous, D. J. and Bandyopadhyay, A. (2005). A survey of max-type recursive distributional equations. Ann. Appl. Probab. 15 1047–1110.
  • [6] Arguin, L.-P. and Chatterjee, S. (2013). Random overlap structures: Properties and applications to spin glasses. Probab. Theory Related Fields 156 375–413.
  • [7] Baik, J., Liechty, K. and Schehr, G. (2012). On the joint distribution of the maximum and its position of the $\mathrm{Airy}_{2}$ process minus a parabola. J. Math. Phys. 53 083303.
  • [8] Bakhtin, Y. and Seo, D. Localization of directed polymers in continuous space. Preprint. Available at arXiv:1905.00930.
  • [9] Barral, J., Rhodes, R. and Vargas, V. (2012). Limiting laws of supercritical branching random walks. C. R. Math. Acad. Sci. Paris 350 535–538.
  • [10] Barraquand, G. and Corwin, I. (2017). Random-walk in beta-distributed random environment. Probab. Theory Related Fields 167 1057–1116.
  • [11] Bates, E. and Chatterjee, S. (2020). Supplement to “The endpoint distribution of directed polymers.” https://doi.org/10.1214/19-AOP1376SUPPA, https://doi.org/10.1214/19-AOP1376SUPPB.
  • [12] Bezerra, S., Tindel, S. and Viens, F. (2008). Superdiffusivity for a Brownian polymer in a continuous Gaussian environment. Ann. Probab. 36 1642–1675.
  • [13] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York.
  • [14] Bolthausen, E. (1989). A note on the diffusion of directed polymers in a random environment. Comm. Math. Phys. 123 529–534.
  • [15] Bolthausen, E., König, W. and Mukherjee, C. (2017). Mean-field interaction of Brownian occupation measures II: A rigorous construction of the Pekar process. Comm. Pure Appl. Math. 70 1598–1629.
  • [16] Borodin, A. and Corwin, I. (2014). Macdonald processes. Probab. Theory Related Fields 158 225–400.
  • [17] Borodin, A., Corwin, I. and Ferrari, P. (2014). Free energy fluctuations for directed polymers in random media in $1+1$ dimension. Comm. Pure Appl. Math. 67 1129–1214.
  • [18] Borodin, A., Corwin, I. and Remenik, D. (2013). Log-gamma polymer free energy fluctuations via a Fredholm determinant identity. Comm. Math. Phys. 324 215–232.
  • [19] Borodin, A. and Petrov, L. (2014). Integrable probability: From representation theory to Macdonald processes. Probab. Surv. 11 1–58.
  • [20] Bothner, T. and Liechty, K. (2013). Tail decay for the distribution of the endpoint of a directed polymer. Nonlinearity 26 1449–1472.
  • [21] Bröker, Y. and Mukherjee, C. (2018). Localization of the Gaussian multiplicative chaos in the Wiener space and the stochastic heat equation in strong disorder. Preprint. Available at arXiv:1808.05202.
  • [22] Burkholder, D. L., Davis, B. J. and Gundy, R. F. (1972). Integral inequalities for convex functions of operators on martingales. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability Theory 223–240.
  • [23] Carmona, P. and Hu, Y. (2002). On the partition function of a directed polymer in a Gaussian random environment. Probab. Theory Related Fields 124 431–457.
  • [24] Carmona, P. and Hu, Y. (2006). Strong disorder implies strong localization for directed polymers in a random environment. ALEA Lat. Am. J. Probab. Math. Stat. 2 217–229.
  • [25] Comets, F. (2017). Directed Polymers in Random Environments. Lecture Notes in Math. 2175. Springer, Cham.
  • [26] Comets, F. and Dembo, A. (2001). Ordered overlaps in disordered mean-field models. Probab. Theory Related Fields 121 1–29.
  • [27] Comets, F. and Nguyen, V.-L. (2016). Localization in log-gamma polymers with boundaries. Probab. Theory Related Fields 166 429–461.
  • [28] Comets, F., Shiga, T. and Yoshida, N. (2003). Directed polymers in a random environment: Path localization and strong disorder. Bernoulli 9 705–723.
  • [29] Comets, F., Shiga, T. and Yoshida, N. (2004). Probabilistic analysis of directed polymers in a random environment: A review. In Stochastic Analysis on Large Scale Interacting Systems. Adv. Stud. Pure Math. 39 115–142. Math. Soc. Japan, Tokyo.
  • [30] Comets, F. and Vargas, V. (2006). Majorizing multiplicative cascades for directed polymers in random media. ALEA Lat. Am. J. Probab. Math. Stat. 2 267–277.
  • [31] Comets, F. and Yoshida, N. (2004). Some new results on Brownian directed polymers in random environment. RIMS Kokyuroku 1386 50–66.
  • [32] Comets, F. and Yoshida, N. (2005). Brownian directed polymers in random environment. Comm. Math. Phys. 254 257–287.
  • [33] Comets, F. and Yoshida, N. (2006). Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 1746–1770.
  • [34] Corwin, I. (2012). The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 1 1130001.
  • [35] Corwin, I., O’Connell, N., Seppäläinen, T. and Zygouras, N. (2014). Tropical combinatorics and Whittaker functions. Duke Math. J. 163 513–563.
  • [36] Corwin, I., Seppäläinen, T. and Shen, H. (2015). The strict-weak lattice polymer. J. Stat. Phys. 160 1027–1053.
  • [37] den Hollander, F. (2009). Random Polymers. Lecture Notes in Math. 1974. Springer, Berlin.
  • [38] Derrida, B. and Spohn, H. (1988). Polymers on disordered trees, spin glasses, and traveling waves. J. Stat. Phys. 51 817–840.
  • [39] Gantert, N., Peres, Y. and Shi, Z. (2010). The infinite valley for a recurrent random walk in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 46 525–536.
  • [40] Georgiou, N., Rassoul-Agha, F. and Seppäläinen, T. (2016). Variational formulas and cocycle solutions for directed polymer and percolation models. Comm. Math. Phys. 346 741–779.
  • [41] Georgiou, N., Rassoul-Agha, F., Seppäläinen, T. and Yilmaz, A. (2015). Ratios of partition functions for the log-gamma polymer. Ann. Probab. 43 2282–2331.
  • [42] Georgiou, N. and Seppäläinen, T. (2013). Large deviation rate functions for the partition function in a log-gamma distributed random potential. Ann. Probab. 41 4248–4286.
  • [43] Golosov, A. O. (1984). Localization of random walks in one-dimensional random environments. Comm. Math. Phys. 92 491–506.
  • [44] Huse, D. A. and Henley, C. L. (1985). Pinning and roughening of domain walls in Ising systems due to random impurities. Phys. Rev. Lett. 54 2708–2711.
  • [45] Imbrie, J. Z. and Spencer, T. (1988). Diffusion of directed polymers in a random environment. J. Stat. Phys. 52 609–626.
  • [46] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476.
  • [47] Johansson, K. (2003). Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 277–329.
  • [48] Kahane, J.-P. and Peyrière, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22 131–145.
  • [49] Kantorovitch, L. (1942). On the translocation of masses. C. R. (Dokl.) Acad. Sci. URSS 37 199–201.
  • [50] Kifer, Y. (1997). The Burgers equation with a random force and a general model for directed polymers in random environments. Probab. Theory Related Fields 108 29–65.
  • [51] König, W. and Mukherjee, C. (2017). Mean-field interaction of Brownian occupation measures, I: Uniform tube property of the Coulomb functional. Ann. Inst. Henri Poincaré Probab. Stat. 53 2214–2228.
  • [52] Lacoin, H. (2010). New bounds for the free energy of directed polymers in dimension $1+1$ and $1+2$. Comm. Math. Phys. 294 471–503.
  • [53] Lévy, P. (1954). Théorie de L’addition des Variables Aléatoires. Gauthier-Villars, Paris.
  • [54] Lions, P.-L. (1984). The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 109–145.
  • [55] Lions, P.-L. (1984). The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 223–283.
  • [56] Lions, P.-L. (1985). The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoam. 1 145–201.
  • [57] Lions, P.-L. (1985). The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoam. 1 45–121.
  • [58] Mejane, O. (2004). Upper bound of a volume exponent for directed polymers in a random environment. Ann. Inst. Henri Poincaré Probab. Stat. 40 299–308.
  • [59] Moreno Flores, G., Quastel, J. and Remenik, D. (2013). Endpoint distribution of directed polymers in $1+1$ dimensions. Comm. Math. Phys. 317 363–380.
  • [60] Mukherjee, C. (2017). Gibbs measures on mutually interacting Brownian paths under singularities. Comm. Pure Appl. Math. 70 2366–2404.
  • [61] Mukherjee, C. and Varadhan, S. R. S. (2016). Brownian occupation measures, compactness and large deviations. Ann. Probab. 44 3934–3964.
  • [62] Munkres, J. R. (2000). Topology. Prentice Hall, Upper Saddle River, NJ.
  • [63] O’Connell, N. (2003). Random matrices, non-colliding processes and queues. In Séminaire de Probabilités, XXXVI. Lecture Notes in Math. 1801 165–182. Springer, Berlin.
  • [64] O’Connell, N. and Yor, M. (2001). Brownian analogues of Burke’s theorem. Stochastic Process. Appl. 96 285–304.
  • [65] Panchenko, D. (2013). The Sherrington–Kirkpatrick Model. Springer Monographs in Mathematics. Springer, New York.
  • [66] Parthasarathy, K. R., Ranga Rao, R. and Varadhan, S. R. S. (1962). On the category of indecomposable distributions on topological groups. Trans. Amer. Math. Soc. 102 200–217.
  • [67] Petermann, M. (2000). Superdiffusivity of polymers in random environment. Ph.D. thesis, Univ. Zürich.
  • [68] Piza, M. S. T. (1997). Directed polymers in a random environment: Some results on fluctuations. J. Stat. Phys. 89 581–603.
  • [69] Quastel, J. and Remenik, D. (2014). Airy processes and variational problems. In Topics in Percolative and Disordered Systems. Springer Proc. Math. Stat. 69 121–171. Springer, New York.
  • [70] Quastel, J. and Remenik, D. (2015). Tails of the endpoint distribution of directed polymers. Ann. Inst. Henri Poincaré Probab. Stat. 51 1–17.
  • [71] Rassoul-Agha, F., Seppäläinen, T. and Yilmaz, A. (2013). Quenched free energy and large deviations for random walks in random potentials. Comm. Pure Appl. Math. 66 202–244.
  • [72] Rassoul-Agha, F., Seppäläinen, T. and Yilmaz, A. (2017). Variational formulas and disorder regimes of random walks in random potentials. Bernoulli 23 405–431.
  • [73] Ruzmaikina, A. and Aizenman, M. (2005). Characterization of invariant measures at the leading edge for competing particle systems. Ann. Probab. 33 82–113.
  • [74] Schehr, G. (2012). Extremes of $N$ vicious walkers for large $N$: Application to the directed polymer and KPZ interfaces. J. Stat. Phys. 149 385–410.
  • [75] Seppäläinen, T. (2012). Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab. 40 19–73.
  • [76] Seppäläinen, T. and Valkó, B. (2010). Bounds for scaling exponents for a $1+1$ dimensional directed polymer in a Brownian environment. ALEA Lat. Am. J. Probab. Math. Stat. 7 451–476.
  • [77] Sinai, Y. G. (1995). A remark concerning random walks with random potentials. Fund. Math. 147 173–180.
  • [78] Sinaĭ, Ya. G. (1982). The limit behavior of a one-dimensional random walk in a random environment. Teor. Veroyatn. Primen. 27 247–258.
  • [79] Song, R. and Zhou, X. Y. (1996). A remark on diffusion of directed polymers in random environments. J. Stat. Phys. 85 277–289.
  • [80] Talagrand, M. (2006). The Parisi formula. Ann. of Math. (2) 163 221–263.
  • [81] Tao, T. (2008). Concentration compactness and the profile decomposition. What’s New (blog Entry). Available at https://terrytao.wordpress.com/2008/11/05/concentration-compactness-and-the-profile-decomposition/.
  • [82] Tao, T. (2010). Concentration compactness via nonstandard analysis. What’s New (blog Entry). Available at https://terrytao.wordpress.com/2010/11/29/concentration-compactness-via-nonstandard-analysis/.
  • [83] Thiery, T. and Le Doussal, P. (2015). On integrable directed polymer models on the square lattice. J. Phys. A 48 465001.
  • [84] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer Series in Statistics. Springer, New York.
  • [85] Vargas, V. (2007). Strong localization and macroscopic atoms for directed polymers. Probab. Theory Related Fields 138 391–410.
  • [86] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin.
  • [87] Wüthrich, M. V. (1998). Superdiffusive behavior of two-dimensional Brownian motion in a Poissonian potential. Ann. Probab. 26 1000–1015.
  • [88] Yoshida, N. (2008). Phase transitions for the growth rate of linear stochastic evolutions. J. Stat. Phys. 133 1033–1058.

Supplemental materials

  • Appendix A: Remaining technical details. This appendix contains the proofs of Proposition 2.4, Theorem 2.8, Proposition 3.4, Lemma 6.1(a), measurability of the support number, Lemma 7.1, and the equivalence of two notions of asymptotic pure atomicity.
  • Appendix B: Comparison to the Mukherjee–Varadhan topology. This appendix proves that the topology introduced by Mukherjee and Varadhan [61], when adapted to the discrete setting, is equivalent to the one constructed in this manuscript.