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Modulus of continuity for polymer fluctuations and weight profiles in Poissonian last passage percolation. (English) Zbl 1439.82014
Summary: In last passage percolation models, the energy of a path is maximized over all directed paths with given endpoints in a random environment, and the maximizing paths are called geodesics. The geodesics and their energy can be scaled so that transformed geodesics cross unit distance and have fluctuations and scaled energy of unit order. Here we consider Poissonian last passage percolation, a model lying in the KPZ universality class, and refer to scaled geodesics as polymers and their scaled energies as weights. Polymers may be viewed as random functions of the vertical coordinate and, when they are, we show that they have modulus of continuity whose order is at most \(t^{2/3}\big(\log t^{-1}\big)^{1/3}\). The power of one-third in the logarithm may be expected to be sharp and in a related problem we show that it is: among polymers in the unit box whose endpoints have vertical separation \(t\) (and a horizontal separation of the same order), the maximum transversal fluctuation has order \(t^{2/3}\big(\log t^{-1}\big)^{1/3}\). Regarding the orthogonal direction, in which growth occurs, we show that, when one endpoint of the polymer is fixed at \((0,0)\) and the other is varied vertically over \((0,z)\), \(z\in [1,2]\), the resulting random weight profile has sharp modulus of continuity of order \(t^{1/3}\big(\log t^{-1}\big)^{2/3}\). In this way, we identify exponent pairs of \((2/3,1/3)\) and \((1/3,2/3)\) in power law and polylogarithmic correction, respectively for polymer fluctuation, and polymer weight under vertical endpoint perturbation. The two exponent pairs describe [A. Hammond, Commun. Math. Phys. 310, No. 2, 455–509 (2012; Zbl 1242.82016); Ann. Probab. 40, No. 3, 921–978 (2012; Zbl 1271.60021); J. Stat. Phys. 142, No. 2, 229–276 (2011; Zbl 1209.82018)] the fluctuation of the boundary separating two phases in subcritical planar random cluster models.
MSC:
82B23 Exactly solvable models; Bethe ansatz
82C23 Exactly solvable dynamic models in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C43 Time-dependent percolation in statistical mechanics
82D60 Statistical mechanical studies of polymers
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References:
[1] Kenneth S. Alexander, Cube-root boundary fluctuations for droplets in random cluster models, Comm. Math. Phys. 224 (2001), no. 3, 733-781. · Zbl 0996.82034
[2] Jinho Baik, Percy Deift, and Kurt Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc 12 (1999), 1119-1178. · Zbl 0932.05001
[3] Riddhipratim Basu, Sourav Sarkar, and Allan Sly, Invariant measures for TASEP with a slow bond, arXiv:1704.07799 (2017).
[4] Riddhipratim Basu, Sourav Sarkar, and Allan Sly, Coalescence of geodesics in exactly solvable models of last passage percolation, J. Math. Phys. 60 (2019), no. 9, 093301, 22. · Zbl 07116349
[5] Riddhipratim Basu, Vladas Sidoravicius, and Allan Sly, Last passage percolation with a defect line and the solution of the slow bond problem, arXiv:1408.3464 (2014). · Zbl 1404.60144
[6] Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. · Zbl 0944.60003
[7] Richard Durrett, Probability: theory and examples, fourth ed., Cambridge Series in Statistical and Probabilistic Mathematics, vol. 31, Cambridge University Press, Cambridge, 2010.
[8] Alan Hammond, Phase separation in random cluster models III: circuit regularity, J. Stat. Phys. 142 (2011), no. 2, 229-276. · Zbl 1209.82018
[9] Alan Hammond, Phase separation in random cluster models I: uniform upper bounds on local deviation, Comm. Math. Phys. 310 (2012), no. 2, 455-509. · Zbl 1242.82016
[10] Alan Hammond, Phase separation in random cluster models II: the droplet at equilibrium, and local deviation lower bounds, Ann. Probab. 40 (2012), no. 3, 921-978. · Zbl 1271.60021
[11] Alan Hammond, Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation, arXiv:1609.02971v3 (2016).
[12] Alan Hammond, Modulus of continuity of polymer weight profiles in Brownian last passage percolation, arXiv:1709.04115v1 (2017).
[13] Kurt Johansson, Transversal fluctuations for increasing subsequences on the plane, Probab. Theory Related Fields 116 (2000), no. 4, 445-456. · Zbl 0960.60097
[14] Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889-892. · Zbl 1101.82329
[15] Matthias Löwe and Franz Merkl, Moderate deviations for longest increasing subsequences: the upper tail., Comm. Pure Appl. Math. 54 (2001), 1488-1519. · Zbl 1033.60035
[16] Matthias Löwe, Franz Merkl, and Silke Rolles, Moderate deviations for longest increasing subsequences: the lower tail., J. Theor. Probab. 15 (2002), no. 4, 1031-1047. · Zbl 1011.60007
[17] Peter Mörters and Yuval Peres, Brownian motion, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 30, Cambridge University Press, Cambridge, 2010, With an appendix by Oded Schramm and Wendelin Werner. · Zbl 1243.60002
[18] Hasan B. Uzun and Kenneth S. Alexander, Lower bounds for boundary roughness for droplets in Bernoulli percolation, Probab. Theory Related Fields 127 (2003), no. 1, 62-88. · Zbl 1034.60090
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