Related papers: Multiple geodesics with the same direction

We consider first-passage percolation on $\mathbb Z^2$ with independent and identically distributed weights whose common distribution is absolutely continuous with a finite exponential moment. Under the assumption that the limit shape has…

We consider a wide class of ergodic first passage percolation processes on Z^2 and prove that there exist at least four one-sided geodesics a.s. We also show that coexistence is possible with positive probability in a four color…

In first-passage percolation, one places nonnegative i.i.d. random variables (T (e)) on the edges of Z d. A geodesic is an optimal path for the passage times T (e). Consider a local property of the time environment. We call it a pattern. We…

We develop a new probabilistic method for deriving deviation estimates in directed planar polymer and percolation models. The key estimates are for exit points of geodesics as they cross transversal down-right boundaries. These bounds are…

The paper shows that the curvature of RP2 is constant iff all geodesics are closed. Therefore RP2 is the first known manifold with only one G-structure. It took quiete a long time to find such a manifold. The author shows only that if all…

This monograph resolves - in a dense class of cases - several open problems concerning geodesics in i.i.d. first-passage percolation on $\mathbb{Z}^d$. Our primary interest is in the empirical measures of edge-weights observed along…

Riemannian first-passage percolation (FPP) is a continuum model, with a distance function arising from a random Riemannian metric in $\R^d$. Our main result is a shape theorem for this model, which says that large balls under this metric…

By recent works of B\"aumler [2] and of the authors of this paper [5], the (limiting) random metric for the critical long-range percolation was constructed. In this paper, we prove the uniqueness of the geodesic between two fixed points,…

Corner percolation is a dependent bond percolation model on Z^2 introduced by B\'alint T\'oth, in which each vertex has exactly two incident edges, perpendicular to each other. G\'abor Pete has proven in 2008 that under the maximal entropy…

If all prime closed geodesics on $(S^n,F)$ with an irreversible Finsler metric $F$ are irrationally elliptic, there exist either exactly $2\left[\frac{n+1}{2}\right]$ or infinitely many distinct closed geodesics. As an application, we show…

In this note we show that for any hyperbolic surface S, the number of geodesics of length bounded above by L in the mapping class group orbit of a fixed closed geodesic with a single double point is asymptotic to L raised to the dimension…

In this note, we study the model of directed last passage percolation on $\mathbb{Z}^2$, with i.i.d. exponential weight. We consider the maximum paths from vertices $\left(0,\lfloor k^{2/3} \rfloor\right)$ and $(\lfloor k^{2/3} \rfloor,0)$…

For exponential last passage percolation on the plane we analyse the probability that the point-to-line geodesic exhibits an atypically large transversal fluctuation at the endpoint as well as the probability that the point-to-point…

In this survey article we consider the directed last-passage percolation model on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside of the class of exactly solvable models. We show how…

We study the problem of coexistence in a two-type competition model governed by first-passage percolation on $\Zd$ or on the infinite cluster in Bernoulli percolation. Actually, we prove for a large class of ergodic stationary passage times…

In the present article we consider a natural generalization of Hammersley's Last Passage Percolation (LPP) called Entropy-controlled Last Passage Percolation (E-LPP), where points can be collected by paths with a global (entropy) constraint…

Since the works of Howard and Newman (2001), it is known that in straight radial rooted trees, with probability 1, infinite paths all have an asymptotic direction and each asymptotic direction is reached by (at least) an infinite path.…

For a geodesic ball with non-negative Ricci curvature and almost maximal volume, without using compactness argument, we construct an $\epsilon$-splitting map on a concentric geodesic ball with uniformly small radius. There are two new…

We show that the number of maximal paths in directed last-passage percolation on the hypercubic lattice ${\mathbb Z}^d$ $(d\geq2)$ in which weights take finitely many values is typically exponentially large.

We establish estimates for the coalescence time of semi-infinite directed geodesics in the planar corner growth model with i.i.d. exponential weights. There are four estimates: upper and lower bounds on the probabilities of both fast and…