Point-to-line last passage percolation and the invariant measure of a system of reflecting Brownian motions
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Point-to-line last passage percolation and the invariant measure of a system of reflecting Brownian motions Will FitzGerald1
· Jon Warren2
Received: 17 July 2019 / Revised: 12 February 2020 © The Author(s) 2020
Abstract This paper proves an equality in law between the invariant measure of a reflected system of Brownian motions and a vector of point-to-line last passage percolation times in a discrete random environment. A consequence describes the distribution of the all-time supremum of Dyson Brownian motion with drift. A finite temperature version relates the point-to-line partition functions of two directed polymers, with an inverse-gamma and a Brownian environment, and generalises Dufresne’s identity. Our proof introduces an interacting system of Brownian motions with an invariant measure given by a field of point-to-line log partition functions for the log-gamma polymer. Keywords Reflected Brownian motions · Random matrices · Dufresne’s identity · Log-gamma polymer · Point-to-line last passage percolation Mathematics Subject Classification 60J65 · 60B20 · 60K35
1 Introduction In this paper we generalise to a random matrix setting the classical identity: d sup B(t) − μt = e(μ)
(1)
t≥0
where B is a Brownian motion, μ > 0 a drift and e(μ) is a random variable which has the exponential distribution with rate 2μ. In our generalisation, the Brownian motion is
B
Will FitzGerald [email protected] Jon Warren [email protected]
1
Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Brighton BN1 9QH, UK
2
Department of Statistics, University of Warwick, Coventry CV4 7AL, UK
123
W. FitzGerald, J. Warren
replaced by the largest eigenvalue process of a Brownian motion with drift on the space of Hermitian matrices (see Sect. 2) and the single exponentially distributed random variable is replaced by a random variable constructed from a field of independent exponentially distributed random variables using the operations of summation and maximum. In fact this latter random variable is well known as a point-to-line last passage percolation time. Theorem 1 Let (H (t) : t ≥ 0) be an n × n Hermitian Brownian motion, let D be an n × n diagonal matrix with entries D j j = α j > 0 for each j = 1, . . . n and let λmax (A) denote the largest eigenvalue of a matrix A. Then d
sup λmax (H (t) − t D) = max t≥0
flat
π ∈Πn (i j)∈π
ei j
where ei j are an independent collection of exponential random variables indexed by N2 ∩ {(i, j) : i + j ≤ n + 1} with rates αi + αn+1− j and the maximum is taken over the set of all directed (up and right) nearest-neighbour paths from (1, 1) to the line flat {(i, j) : i + j = n + 1} which we denote by Πn . This result gives a connection between random matrix theory and the Kardar-ParisiZhang (KPZ) universality class, a collection of models related to random interface growth including growth models, directed polymers in a random environment and various interacting particle systems. Connections of this form originated in the semin
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