Second class particles and cube root asymptotics for Hammersley's process

Longest increasing paths with Lipschitz constraints

The Hammersley problem asks for the maximal number of points in a monotonous path through a Poisson point process. It is exactly solvable and notoriously known to belong to the KPZ universality class, with a cube-root scaling for the fluctuations.Here we introduce and analyze a variant in which we impose a Lipschitz condition on paths. Thanks to a coupling with the classical Hammersley problem we observe that this variant is also exactly solvable. It allows us to derive first and second orders asymptotics. It turns out that the cube-root scaling only holds for certain choices of the Lipschitz …

Cube root fluctuations for the corner growth model associated to the exclusion process

We study the last-passage growth model on the planar integer lattice with exponential weights. With boundary conditions that represent the equilibrium exclusion process as seen from a particle right after its jump we prove that the variance of the last-passage time in a characteristic direction is of order t^{2/3}. With more general boundary conditions that include the rarefaction fan case we show that the last-passage time fluctuations are still of order t^{1/3}, and also that the transversal fluctuations of the maximal path have order t^{2/3}. We adapt and then build on a recent study of Ham…

Scaling limit and cube-root fluctuations in SOS surfaces above a wall

Consider the classical $(2+1)$-dimensional Solid-On-Solid model above a hard wall on an $L\times L$ box of $\bbZ^2$. The model describes a crystal surface by assigning a non-negative integer height $η_x$ to each site $x$ in the box and 0 heights to its boundary. The probability of a surface configuration $η$ is proportional to $\exp(-β\mathcal{H}(η))$, where $β$ is the inverse-temperature and $\mathcal{H}(η)$ sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough temperatures. First we show that with h…

面积约束随机游走模型中的精确立方根涨落

Exact cube-root fluctuations in an area-constrained random walk model

本文研究了面积约束下(1+1)维随机游走的全局凹轨迹局部偏差，引入平均面长度和平均局部粗糙度两个量，证明其涨落阶分别为$N^{2/3}$和$N^{1/3}$，并确定了多对数修正，为二维统计力学相分离界面提供了玩具模型。

Cube-root boundary fluctuations for droplets in random cluster models

For a family of bond percolation models on Z^{2} that includes the Fortuin-Kasteleyn random cluster model, we consider properties of the ``droplet'' that results, in the percolating regime, from conditioning on the existence of an open dual circuit surrounding the origin and enclosing at least (or exactly) a given large area A. This droplet is a close surrogate for the one obtained by Dobrushin, Kotecký and Shlosman by conditioning the Ising model; it approximates an area-A Wulff shape. The local part of the deviation from the Wulff shape (the ``local roughness'') is the inward deviation of th…

Cube root weak convergence of empirical estimators of a density level set

Given $n$ independent random vectors with common density $f$ on $\mathbb{R}^d$, we study the weak convergence of three empirical-measure based estimators of the convex $λ$-level set $L_λ$ of $f$, namely the excess mass set, the minimum volume set and the maximum probability set, all selected from a class of convex sets $\mathcal{A}$ that contains $L_λ$. Since these set-valued estimators approach $L_λ$, even the formulation of their weak convergence is non-standard. We identify the joint limiting distribution of the symmetric difference of $L_λ$ and each of the three estimators, at rate $n^{-1/…