Limiting geodesics for first-passage percolation on subsets of Z2\mathbb{Z}^2

It is an open problem to show that in two-dimensional first-passage percolation, the sequence of finite geodesics from any point to $(n,0)$ has a limit in $n$. In this paper, we consider this question for first-passage percolation on a wide class of subgraphs of $\mathbb {Z}^2$: those whose vertex set is infinite and connected with an infinite connected complement. This includes, for instance, slit planes, half-planes and sectors. Writing $x_n$ for the sequence of boundary vertices, we show that the sequence of geodesics from any point to $x_n$ has an almost sure limit assuming only existence of finite geodesics. For all passage-time configurations, we show existence of a limiting Busemann function. Specializing to the case of the half-plane, we prove that the limiting geodesic graph has one topological end; that is, all its infinite geodesics coalesce, and there are no backward infinite paths. To do this, we prove in the Appendix existence of geodesics for all product measures in our domains and remove the moment assumption of the Wehr-Woo theorem on absence of bigeodesics in the half-plane.Comment: Published in at http://dx.doi.org/10.1214/13-AAP999 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Rate of convergence of the mean for sub-additive ergodic sequences

For sub-additive ergodic processes $\{X_{m,n}\}$ with weak dependence, we analyze the rate of convergence of $\mathbb{E}X_{0,n}/n$ to its limit $g$. We define an exponent $\gamma$ given roughly by $\mathbb{E}X_{0,n} \sim ng + n^\gamma$, and, assuming existence of a fluctuation exponent $\chi$ that gives $\mathrm{Var}~X_{0,n} \sim n^{2\chi}$, we provide a lower bound for $\gamma$ of the form $\gamma \geq \chi$. The main requirement is that $\chi \neq 1/2$. In the case $\chi=1/2$ and under the assumption $\mathrm{Var}~X_{0,n} = O(n/(\log n)^\beta)$ for some $\beta>0$, we prove $\gamma \geq \chi - c(\beta)$ for a $\beta$-dependent constant $c(\beta)$. These results show in particular that non-diffusive fluctuations are associated to non-trivial $\gamma$. Various models, including first-passage percolation, directed polymers, the minimum of a branching random walk and bin packing, fall into our general framework, and the results apply assuming $\chi$ exists. In the case of first-passage percolation in $\mathbb Z^d$, we provide a version of $\gamma \geq -1/2$ without assuming existence of $\chi$.Comment: 41 pages, 2 figures. Expanded introduction and added a proof sketc

50 years of first passage percolation

We celebrate the 50th anniversary of one the most classical models in probability theory. In this survey, we describe the main results of first passage percolation, paying special attention to the recent burst of advances of the past 5 years. The purpose of these notes is twofold. In the first chapters, we give self-contained proofs of seminal results obtained in the '80s and '90s on limit shapes and geodesics, while covering the state of the art of these questions. Second, aside from these classical results, we discuss recent perspectives and directions including (1) the connection between Busemann functions and geodesics, (2) the proof of sublinear variance under 2+log moments of passage times and (3) the role of growth and competition models. We also provide a collection of (old and new) open questions, hoping to solve them before the 100th birthday.Comment: 160 pages, 17 figures. This version has updated chapters 3-5, with expanded and additional material. Small typos corrected throughou