Abstract
In last passage percolation models, the energy of a path is maximized over all directed paths with given endpoints in a random environment, and the maximizing paths are called geodesics. The geodesics and their energy can be scaled so that transformed geodesics cross unit distance and have fluctuations and scaled energy of unit order. Here we consider Poissonian last passage percolation, a model lying in the KPZ universality class, and refer to scaled geodesics as polymers and their scaled energies as weights. Polymers may be viewed as random functions of the vertical coordinate and, when they are, we show that they have modulus of continuity whose order is at most $t^{2/3}\big (\log t^{-1}\big )^{1/3}$. The power of one-third in the logarithm may be expected to be sharp and in a related problem we show that it is: among polymers in the unit box whose endpoints have vertical separation $t$ (and a horizontal separation of the same order), the maximum transversal fluctuation has order $t^{2/3}\big (\log t^{-1}\big )^{1/3}$. Regarding the orthogonal direction, in which growth occurs, we show that, when one endpoint of the polymer is fixed at $(0,0)$ and the other is varied vertically over $(0,z)$, $z\in [1,2]$, the resulting random weight profile has sharp modulus of continuity of order $t^{1/3}\big (\log t^{-1}\big )^{2/3}$. In this way, we identify exponent pairs of $(2/3,1/3)$ and $(1/3,2/3)$ in power law and polylogarithmic correction, respectively for polymer fluctuation, and polymer weight under vertical endpoint perturbation. The two exponent pairs describe [9, 10, 8] the fluctuation of the boundary separating two phases in subcritical planar random cluster models.
Highlights
In last passage percolation models, the energy of a path is maximized over all directed paths with given endpoints in a random environment, and the maximizing paths are called geodesics
Polymer fluctuations and weight profiles in last passage percolation that, while they are reliant on certain integrable inputs, are probabilistic in flavour: for example, [5], [4] and [3] are recent results and applications concerning geometric properties of last passage percolation paths
That last passage percolation paths experience fluctuation in their energy and transversal fluctuation governed by scaling exponents of one-third and two-thirds
Summary
Let Π be a homogeneous rate one Poisson point process (PPP) on R2. We introduce a partial order on R2: (x1, y1) (x2, y2) if and only if x1 ≤ x2 and y1 ≤ y2. We define the last passage time from u to v, denoted by Xuv, to be the maximum of |γ| as γ varies over all increasing paths from u to v. Any such maximizing path is called a geodesic. (The notation ← and → is compatible with these two paths being well described as the leftmost and rightmost geodesics This choice of notation anticipates the form of these paths when viewed in the scaled coordinates that we are about to introduce.) When the endpoints are (0, 0) and (n, n), we will call these geodesics Γ← n and Γ→ n.
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