Regularity of the time constant for a supercritical Bernoulli percolation

First Passage ◽

Supercritical Regime ◽

Regularity Properties ◽

Chemical Distance ◽

Special Case

We consider an i.i.d. supercritical bond percolation on Z^d, every edge is open with a probability p > p_c(d), where p_c(d) denotes the critical parameter for this percolation. We know that there exists almost surely a unique infinite open cluster C_p [11]. We are interested in the regularity properties of the chemical distance for supercritical Bernoulli percolation. The chemical distance between two points x, y ∈ C_p corresponds to the length of the shortest path in C_p joining the two points. The chemical distance between 0 and nx grows asymptotically like nµ_p(x). We aim to study the regularity properties of the map p → µ_p in the supercritical regime. This may be seen as a special case of first passage percolation where the distribution of the passage time is G_p = pδ_1 + (1 − p)δ_∞, p > p_c(d). It is already known that the map p → µ_p is continuous (see [10]).

On the time constant and path length of first-passage percolation

Advances in Applied Probability ◽

10.1017/s0001867800020127 ◽

1980 ◽

Vol 12 (04) ◽

pp. 848-863 ◽

Cited By ~ 6

Author(s):

Harry Kesten

Keyword(s):

Distribution Function ◽

Time Constant ◽

Path Length ◽

Passage Time ◽

Square Lattice ◽

Bond Percolation ◽

Critical Probability ◽

First Passage ◽

Individual Bond

Let U be the distribution function of the passage time of an individual bond of the square lattice, and let pT be the critical probability above which the expected size of the open component of the origin (in the usual bond percolation) is infinite. It is shown that if (∗)U(0–) = 0, U(0) < pT , then there exist constants 0 < a, C 1 < ∞ such that a self-avoiding path of at least n steps starting at the origin and with passage time ≦ an} ≦ 2 exp (–C 1 n). From this it follows that under (∗) the time constant μ (U) of first-passage percolation is strictly positive and that for each c > 0 lim sup (1/n)Nn (c) <∞, where Nn (c) is the maximal number of steps in the paths starting at the origin with passage time at most cn.

On the time constant and path length of first-passage percolation

Advances in Applied Probability ◽

10.2307/1426744 ◽

1980 ◽

Vol 12 (4) ◽

pp. 848-863 ◽

Cited By ~ 19

Author(s):

Harry Kesten

Keyword(s):

Distribution Function ◽

Time Constant ◽

Path Length ◽

Passage Time ◽

Square Lattice ◽

Bond Percolation ◽

Critical Probability ◽

First Passage ◽

Individual Bond

Let U be the distribution function of the passage time of an individual bond of the square lattice, and let pT be the critical probability above which the expected size of the open component of the origin (in the usual bond percolation) is infinite. It is shown that if (∗)U(0–) = 0, U(0) < pT, then there exist constants 0 < a, C1 < ∞ such that a self-avoiding path of at least n steps starting at the origin and with passage time ≦ an} ≦ 2 exp (–C1n).From this it follows that under (∗) the time constant μ (U) of first-passage percolation is strictly positive and that for each c > 0 lim sup (1/n)Nn(c) <∞, where Nn(c) is the maximal number of steps in the paths starting at the origin with passage time at most cn.

On the continuity of the time constant of first-passage percolation

10.2307/3213056 ◽

1981 ◽

Vol 18 (4) ◽

pp. 809-819 ◽

Cited By ~ 16

Author(s):

J. Theodore Cox ◽

Harry Kesten

Keyword(s):

Distribution Function ◽

Time Constant ◽

Passage Time ◽

Square Lattice ◽

First Passage ◽

Finite Constant

Let U be the distribution function of the non-negative passage time of an individual edge of the square lattice, and let a0n be the minimal passage time from (0, 0) to (n, 0). The process a0n/n converges in probability as n → ∞to a finite constant μ (U) called the time constant. It is proven that μ (Uk)→ μ(U) whenever Uk converges weakly to U.

The density of interfaces: a new first-passage problem

10.1017/s0021900200044612 ◽

1993 ◽

Vol 30 (04) ◽

pp. 851-862 ◽

Cited By ~ 1

Author(s):

L. Chayes ◽

C. Winfield

Keyword(s):

Correlation Length ◽

First Passage Time ◽

Passage Time ◽

Scaling Behavior ◽

First Passage ◽

Site Percolation ◽

Percolation Problem ◽

First Passage Problem

We introduce and study a novel type of first-passage percolation problem onwhere the associated first-passage time measures the density of interface between two types of sites. If the types, designated + and –, are independently assigned their values with probabilitypand (1 —p) respectively, we show that the density of interface is non-zero provided that both species are subcritical with regard to percolation, i.e.pc>p> 1 –pc.Furthermore, we show that asp↑pcorp↓ (1 –pc), the interface density vanishes with scaling behavior identical to the correlation length of the site percolation problem.

Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster

ESAIM Probability and Statistics ◽

10.1051/ps:2004009 ◽

2004 ◽

Vol 8 ◽

pp. 169-199 ◽

Cited By ~ 10

Author(s):

Olivier Garet ◽

Régine Marchand

Keyword(s):

First Passage ◽

Asymptotic Shape ◽

Chemical Distance

The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier

10.1239/jap/1032265215 ◽

1998 ◽

Vol 35 (3) ◽

pp. 671-682 ◽

Cited By ~ 42

Author(s):

Anders Martin-Löf

Keyword(s):

Differential Equation ◽

Wiener Process ◽

Critical Parameter ◽

Spectral Representation ◽

First Passage Time ◽

Passage Time ◽

Initial Number ◽

Airy Functions ◽

Final Size ◽

First Passage

The distribution of the final size, K, in a general SIR epidemic model is considered in a situation when the critical parameter λ is close to 1. It is shown that with a ‘critical scaling’ λ ≈ 1 + a / n1/3, m ≈ bn1/3, where n is the initial number of susceptibles and m is the initial number of infected, then K / n2/3 has a limit distribution when n → ∞. It can be described as that of T, the first passage time of a Wiener process to a parabolic barrier b + at − t2/2. The proof is based on a diffusion approximation. Moreover, it is shown that the distribution of T can be expressed analytically in terms of Airy functions using the spectral representation connected with Airy's differential equation.

First-passage percolation on the square lattice. I

Advances in Applied Probability ◽

10.1017/s000186780004310x ◽

1977 ◽

Vol 9 (01) ◽

pp. 38-54 ◽

Cited By ~ 5

Author(s):

R. T. Smythe ◽

John C. Wierman

Keyword(s):

Time Constant ◽

First Passage Time ◽

Passage Time ◽

Square Lattice ◽

Ergodic Theorems ◽

Critical Percolation ◽

First Passage ◽

Underlying Distribution ◽

Percolation Probability

We consider several problems in the theory of first-passage percolation on the two-dimensional integer lattice. Our results include: (i) a mean ergodic theorem for the first-passage time from (0,0) to the line x = n; (ii) a proof that the time constant is zero when the atom at zero of the underlying distribution exceeds C, the critical percolation probability for the square lattice; (iii) a proof of the a.s. existence of routes for the unrestricted first-passage processes; (iv) a.s. and mean ergodic theorems for a class of reach processes; (v) continuity results for the time constant as a functional of the underlying distribution.

On the continuity of the time constant of first-passage percolation

10.1017/s0021900200034161 ◽

1981 ◽

Vol 18 (04) ◽

pp. 809-819

Author(s):

J. Theodore Cox ◽

Harry Kesten

Keyword(s):

Distribution Function ◽

Time Constant ◽

Passage Time ◽

Square Lattice ◽

First Passage ◽

Finite Constant

Lower bounds for point-to-point wandering exponents in Euclidean first-passage percolation

10.1017/s0021900200018222 ◽

2000 ◽

Vol 37 (04) ◽

pp. 1061-1073

Author(s):

C. Douglas Howard

Keyword(s):

Lower Bounds ◽

Upper Bound ◽

Line Segment ◽

Passage Time ◽

Straight Line Segment ◽

First Passage ◽

Straight Line ◽

Point To Point

In first-passage percolation models, the passage time T(0,L) from the origin to a point L is expected to exhibit deviations of order |L|χ from its mean, while minimizing paths are expected to exhibit fluctuations of order |L|ξ away from the straight line segment . Here, for Euclidean models in dimension d, we establish the lower bounds ξ ≥ 1/(d+1) and χ ≥(1-(d-1)ξ)/2. Combining this latter bound with the known upper bound ξ ≤ 3/4 yields that χ ≥ 1/8 for d=2.

Remarks on renewal theory for percolation processes

10.1017/s0021900200094353 ◽

1976 ◽

Vol 13 (02) ◽

pp. 290-300 ◽

Cited By ~ 7

Author(s):

R. T. Smythe

Keyword(s):

Ergodic Theorem ◽

First Passage Time ◽

Renewal Theory ◽

Passage Time ◽

Integer Lattice ◽

Two Dimensional ◽

First Passage ◽

Underlying Distribution ◽

Percolation Processes

We extend some results of Hammersley and Welsh concerning first-passage percolation on the two-dimensional integer lattice. Our results include: (i) weak renewal theorems for the unrestricted reach processes; (ii) an L 1-ergodic theorem for the unrestricted first-passage time from (0, 0) to the line X = n; and (iii) weakening of the boundedness restrictions on the underlying distribution in Hammersley and Welsh's weak renewal theorems for the cylinder reach processes.