Geodesics, bigeodesics, and coalescence in first passage percolation in general dimension
Kenneth S. Alexander
备注：
84 pages, 20 figures. Hypotheses weakened for main theorems (see A2.) Section 2 (paper outline) added. Proofs of technical lemmas/propositions moved to appendices. Table of Notation and figures 2, 4, 13 added
We consider geodesics for first passage percolation (FPP) on $\mathbb{Z}^d$ with iid passage times. As has been common in the literature, we assume that the FPP system satisfies certain basic properties conjectured to be true, and derive consequences from these properties. The assumptions are roughly as follows: (i) the fluctuation scale $σ(r)$ of the passage time on scale $r$ grows approximately as a positive power $r^χ$, in the sense that two natural definitions of $σ(r)$ and $χ$ yield the same value $χ$, and (ii) the limit shape boundary has curvature uniformly bounded away from 0 and $\infty$ (a requirement we can sometimes limit to a neighborhood of some fixed direction.) The main a.s. consequences derived are the following, with $ν$ denoting a subpolynomial function and $ξ=(1+χ)/2$ the transverse wandering exponent: (a) for one-ended geodesic rays with a given asymptotic direction $θ$, starting in a natural halfspace $H$, for the hyperplane at distance $r$ from $H$, the density of "entry points" where some geodesic ray first crosses the hyperplane is at most $ν(r)/r^{(d-1)ξ}$, (b) the system has no bigeodesics, i.e. two-ended infinite geodesics, (c) given two sites $x,y$, and a third site $z$ at distance at least $\ell$ from $x$ and $y$, the probability that the geodesic from $x$ to $y$ passes through $z$ is at most $ν(\ell)/\ell^{(d-1)ξ}$, and (d) in $d=2$, the probability that the geodesic rays in a given direction from two sites have not coalesced after distance $r$ decays like $r^{-ξ}$ to within a subpolynomial factor. Our entry-point density bound compares to a natural conjecture of $c/r^{(d-1)ξ}$, corresponding to a spacing of order $r^ξ$ between entry points, which is the conjectured scale of the transverse wandering.
It is believed that, under very general conditions, bi-infinite geodesics (or bigeodesics) do not exist for planar first and last passage percolation (LPP) models. However, if one endows the model with a natural dynamics, thereby gradually perturbing the geometry, then it is plausible that there could exist a non-trivial set $\mathscr{T}$ of exceptional times at which such bigeodesics exist. For dynamical exponential LPP, we show that $\mathscr{T}$ is "very close" to being non-trivial; namely, we obtain an $Ω( 1/\log n)$ lower bound on the probability that there exists a random time $t\in [0,1…
Bi-infinite geodesics are fundamental objects of interest in planar first passage percolation. A longstanding conjecture states that under mild conditions there are almost surely no bigeodesics, however the result has not been proved in any case. For the exactly solvable model of directed last passage percolation on $\mathbb{Z}^2$ with i.i.d. exponential passage times, we study the corresponding question and show that almost surely the only bigeodesics are the trivial ones, i.e., the horizontal and vertical lines. The proof makes use of estimates for last passage time available from the integr…
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 $(\lfloor k^{2/3} \rfloor,0)$ to $(n,n)$, respectively. For the coalescing point of these paths, we show that the probability for it being $>kR$ far away from the origin is in the order of $R^{-2/3}$. This is motivated by a recent work of Basu, Sarkar, and Sly, where the same estimate was obtained for semi-infinite geodesics, and the optimal exponent for the finite case was left open. Ou…
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 slow coalescence on the correct spatial scale with exponent $3/2$. Our proofs utilize a geodesic duality introduced by Pimentel and properties of the increment-stationary last-passage percolation process. For fast coalescence our bounds are new and they have matching optimal exponential order of magnitude. For slow coalescence we reproduce bounds proved earlie…
We prove the existence of semi-infinite geodesics for Brownian last-passage percolation (BLPP). Specifically, on a single event of probability one, there exist semi-infinite geodesics started from every space-time point and traveling in every asymptotic direction. Properties of these geodesics include uniqueness for a fixed initial point and direction, non-uniqueness for fixed direction but random initial points, and coalescence of all geodesics traveling in a common, fixed direction. Along the way, we prove that for fixed northeast and southwest directions, there almost surely exist no bi-inf…
We report on the exact treatment of a random-matrix representation of bond percolation model on a square lattice in two dimensions with occupation probability $p$. The percolation problem is mapped onto a random complex matrix composed of two random real-valued matrices of elements $+1$ and $-1$ with probability $p$ and $1-p$, respectively. We find that the onset of percolation transition can be detected by the emergence of power-law divergences due to the coalescence of the first two extreme eigenvalues in the thermodynamic limit. We develop a universal finite-size scaling law that fully char…
The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the Airy$_2$ process, describes after the subtraction of a parabola the limiting law of the scaled energy of a geodesic running from the origin to a variable point on an anti-diagonal line in such problems as Poissonian last passage percolation. The ensemble of curves resulting from the Airy line ensemble after the subtraction of the same para…
Geodesic Trees and Exceptional Directions in FPP on Hyperbolic Groups
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（国际理论科学中心，塔塔基础研究所）；School of Mathematics, Tata Institute of Fundamental Research（数学学院，塔塔基础研究所）
