Long and Short Time Asymptotics of the Two-Time Distribution in Local Random Growth
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Long and Short Time Asymptotics of the Two-Time Distribution in Local Random Growth Kurt Johansson1 Received: 23 June 2020 / Accepted: 26 October 2020 / © The Author(s) 2020
Abstract The two-time distribution gives the limiting joint distribution of the heights at two different times of a local 1D random growth model in the curved geometry. This distribution has been computed in a specific model but is expected to be universal in the KPZ universality class. Its marginals are the GUE Tracy-Widom distribution. In this paper we study two limits of the two-time distribution. The first, is the limit of long time separation when the quotient of the two times goes to infinity, and the second, is the short time limit when the quotient goes to zero. Keywords Random growth · Two-time · Random matrices
1 Introduction and Results 1.1 Introduction In this paper we will consider the short and long time separation limits of the asymptotic two-time distribution in a polynuclear growth model or, equivalently in a directed last-passage percolation model. For background on these models which belong to the so called KPZ universality class, we refer to [3] and [22]. Let us recall the result on the two-time distribution from [18]. Let (w(i, j ))i,j 1 be independent geometric random variables with parameter q, P[w(i, j ) = k] = (1 − q)q k ,
k 0.
Supported by the grant KAW 2015.0270 from the Knut and Alice Wallenberg Foundation and grant 2015-04872 from the Swedish Science Research Council (VR) Kurt Johansson
[email protected] 1
Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden
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Math Phys Anal Geom
Consider the last-passage times G(m, n) =
max
π:(1,1)(m,n)
(2020) 23:43
w(i, j ),
(1)
(i,j )∈π
where the maximum is over all up/right paths from (1, 1) to (m, n), see [15]. It follows from (1) that we have stochastic recursion formula G(m, n) = max(G(m − 1, n), G(m, n − 1)) + w(m, n).
(2)
We can relate this to a random growth model with an evolving height function as follows. Let G(m, n) = 0 if (m, n) ∈ / Z2+ , and define the height function h(x, t) by t +x+1 t −x+1 h(x, t) = G , (3) 2 2 for x + t odd, and extend it to all x ∈ R by linear interpolation. Then (2) leads to a growth rule for h(x, t) and this is the discrete time and space polynuclear growth model. We think of x → h(x, t) as the height above x at time t, and we get a random one-dimensional interface. Let the constants ci be given by √ √ 2 q q 1/6 (1 + q)1/3 √ 2/3 −1/6 c1 = q (1 + q) , c2 = . (4) √ , c3 = √ 1− q 1− q Consider the rescaled height function HT (η, t) =
h(2c1 η(tT )2/3 , 2tT ) − c2 tT , c3 (tT )1/3
(5)
as a process in η ∈ R and t > 0. It follows from [16] that for a fixed t > 0, the process converges, as T → ∞, to A2 (η) − η2 , where A2 (η) is the Airy-2-process. In particular, for any fixed η, t, lim P[HT (η, t) ξ − η2 ] = F2 (ξ ) = det(I − KAi )L2 (ξ,∞) ,
T →∞
where F2 is the Tracy-Widom distribution, and ∞ Ai (x + s)Ai (y + s) ds, KAi (x, y) =
(6)
0
is the Airy kernel. It
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