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A patchwork quilt sewn from Brownian fabric: regularity of polymer weight profiles in Brownian last passage percolation. (English) Zbl 1427.82036
In percolation models that belong to the Kardar-Paris-Zhang (KPZ) class, the study of long energy-maximizing paths behavior can be shown with the use of the function of the paths’ pair of endpoint locations. The paper considers Brownian last passage percolation in scaled coordinates crossing unit distances with unit-order fluctuations. Very interesting is the case of polymers, when one endpoint is fixed at $$(0, 0) \in \mathbb{R}^2$$ and the other point is given as $$(z, 1)$$, $$z \in \mathbb{R}$$, with polymer weight profile represented by a locally Brownian function of $$z \in \mathbb{R}$$. The object profile can be compared to a Brownian bridge on a given compact interval, with a Radon-Nikodým derivative in every $$L^p$$ space for $$p \in (1,\infty)$$. This paper generalizes some previous exiting results: polymer weight profiles can have any general initial condition; after the affine adjustment the profile can have a Radon-Nikodým derivative that is in the $$L^p$$ space for $$p\in (1, 3)$$.
The whole paper consists of nine sections and two appendices. All required preliminaries are given in the introduction with Theorem 1.2 as a main result. Then, in Section 2, we have a detailed explanation of the geometric meaning of this theorem. Section 3 shows how Theorem 1.2 can be proved. However, some additional details are also given in Section 4. This concept of rerouting is described and proposed in Section 5 as the main key to the proof. Section 6 introduces the problem of late coalescence events. This section, together with the concept of rerouting, is used in Section 7 where we have a detailed description of polymer coalescence, canopies and intra-canopy weight profiles. The final Sections 8 and 9 give the proof with all necessary calculations.

##### MSC:
 82C43 Time-dependent percolation in statistical mechanics 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C22 Interacting particle systems in time-dependent statistical mechanics 82B23 Exactly solvable models; Bethe ansatz 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 82D60 Statistical mechanical studies of polymers
##### Keywords:
polymers; Brownian percolation
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##### References:
 [1] Baik, J.; Deift, P.; Johansson, K., On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc., 12, 4, 1119-1178, (1999) · Zbl 0932.05001 [2] Baryshnikov, Y., GUEs and queues, Probab. Theory Related Fields, 119, 2, 256-274, (2001) · Zbl 0980.60042 [3] Basu, R.; Sarkar, S.; Sly, A. [4] Basu, R.; Sidoravicius, V.; Sly, A. [5] Borodin, A.; Ferrari, P. L.; Prähofer, M.; Sasamoto, T., Fluctuation properties of the TASEP with periodic initial configuration, J. Stat. Phys., 129, 5-6, 1055-1080, (2007) · Zbl 1136.82028 [6] Cator, E.; Pimentel, L. P. R., On the local fluctuations of last-passage percolation models, Stoch. Process. Appl., 125, 2, 538-551, (2015) · Zbl 1326.60134 [7] Corwin, I., The Kardar-Parisi-Zhang equation and universality class, Random Matrices Theory Appl., 1, 1, (2012) · Zbl 1247.82040 [8] Corwin, I.; Hammond, A., Brownian Gibbs property for Airy line ensembles, Invent. Math., 195, 2, 441-508, (2014) · Zbl 06261669 [9] Corwin, I.; Quastel, J.; Remenik, D., Renormalization fixed point of the KPZ universality class, J. Stat. Phys., 160, 4, 815-834, (2015) · Zbl 1327.82064 [10] Dauvergne, D.; Ortmann, J.; Virág, B. [11] Dauvergne, D.; Virág, B. [12] Eden, M., A two-dimensional growth process, Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. IV, 223-239, (1961), University of California Press: University of California Press, Berkeley, CA [13] Glynn, P. W.; Whitt, W., Departures from many queues in series, Ann. Appl. Probab., 1, 4, 546-572, (1991) · Zbl 0749.60090 [14] Gravner, J.; Tracy, C. A.; Widom, H., Limit theorems for height fluctuations in a class of discrete space and time growth models, J. Stat. Phys., 102, 5-6, 1085-1132, (2001) · Zbl 0989.82030 [15] Hägg, J., Local Gaussian fluctuations in the Airy and discrete PNG processes, Ann. Probab., 36, 3, 1059-1092, (2008) · Zbl 1142.60025 [16] Hammond, A. [17] Hammond, A. [18] Hammond, A. [19] Johansson, K., Transversal fluctuations for increasing subsequences on the plane, Probab. Theory Related Fields, 116, 4, 445-456, (2000) · Zbl 0960.60097 [20] Johansson, K., Discrete polynuclear growth and determinantal processes, Comm. Math. Phys., 242, 1-2, 277-329, (2003) · Zbl 1031.60084 [21] Matetski, K.; Quastel, J.; Remenik, D. [22] Moreno Flores, G.; Quastel, J.; Remenik, D., Endpoint distribution of directed polymers in 1 + 1 dimensions, Comm. Math. Phys., 317, 2, 363-380, (2013) · Zbl 1257.82117 [23] O’Connell, N.; Yor, M., A representation for non-colliding random walks, Electron. Comm. Probab., 7, 1-12, (2002) · Zbl 1037.15019 [24] Pimentel, L. P. R., On the location of the maximum of a continuous stochastic process, J. Appl. Probab., 51, 1, 152-161, (2014) · Zbl 1305.60029 [25] Pimentel, L. [26] Prähofer, M.; Spohn, H., Scale invariance of the PNG droplet and the Airy process, J. Stat. Phys., 108, 5-6, 1071-1106, (2002) · Zbl 1025.82010 [27] Quastel, J.; Remenik, D., Local behavior and hitting probabilities of the Airy_1 process, Probab. Theory Related Fields, 157, 3-4, 605-634, (2013) · Zbl 1285.60095 [28] Sasamoto, T., Spatial correlations of the 1D KPZ surface on a flat substrate, J. Phys. A, 38, 33, L549-L556, (2005)
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