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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
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