Intermediate Disorder Regime for Half-Space Directed Polymers
- PDF / 561,751 Bytes
- 32 Pages / 439.37 x 666.142 pts Page_size
- 20 Downloads / 187 Views
Intermediate Disorder Regime for Half-Space Directed Polymers Xuan Wu1 Received: 1 September 2020 / Accepted: 27 October 2020 / Published online: 17 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We consider the convergence of point-to-point partition functions for the half-space directed polymer model in dimension 1+1 in the intermediate disorder regime as introduced for the full space model by Alberts, Khanin and Quastel in [1]. By scaling the inverse temperature as βn −1/4 , the point-to-point partition function converges to the chaos series for the solution to stochastic heat equation with Robin boundary condition and delta initial data. Furthermore, the convergence result is then applied to the exact-solvable log-gamma directed polymer model in a half-space.
1 Introduction The directed polymers were introduced in the statistical physics literature by Huse and Henley [14] in 1985 and received first rigorous mathematical treatment in 1988 by Imbrie and Spencer [15]. The monograph [6] is a great resource for the foundational work in this area. Over the last thirty years, the directed polymers played an important role as a playground of many fascinating problems in the probability world. Among those different directions opened up by directed polymers, in dimension 1+1, its connection to the KPZ universality class [7] has attracted extensive attention. The polymer measure in dimension 1+1 is a random probability measure on paths in a random environment, which favors higher weighted paths. It is constructed through up / right paths on Z2 with path measure re-weighted by an i.i.d. random environment presented at each lattice points. The KPZ universality conjecture concerns the large scale asymptotic behavior of the polymer free energy and there are two characteristic scalings , the 1:2:3 KPZ scaling and the weak noise scaling, known as the strong KPZ universality conjecture and the weak KPZ universality conjecture respectively. In the direction of the strong KPZ universality conjecture for directed polymers, the first rigorous verification of the 1/3 fluctuation of polymer free energy was proven for a special case
Communicated by Eric A. Carlen.
B 1
Xuan Wu [email protected] Department of Mathematics, University of Chicago, Eckhart Hall, 5734 S University Ave, Chicago, IL 60637, USA
123
Intermediate Disorder Regime for Half-Space Directed Polymers
2373
[20], where the integrable log-gamma polymers were introduced. Among directed polymers, the log-gamma directed polymer model was special in the same way as the last passage percolation models with exponential or geometric weights are special among corner growth models. Namely, both demonstrate integrable structures and permit explicit computations. [8] computed the Laplace transform of the point-to-point partition function. [5] transformed that formula into a Fredholm determinant and performed asymptotic analysis, with motivation from Macdonald process formulas in [4]. Under the weak noise scaling, the convergence o
Data Loading...
