The density of interfaces: a new first-passage problem

1993 ◽  
Vol 30 (04) ◽  
pp. 851-862 ◽  
Author(s):  
L. Chayes ◽  
C. Winfield
Keyword(s):  
Correlation Length ◽  
First Passage Time ◽  
Passage Time ◽  
Scaling Behavior ◽  
First Passage Percolation ◽  
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.

The density of interfaces: a new first-passage problem

1993 ◽  
Vol 30 (4) ◽  
pp. 851-862 ◽  
Author(s):  
L. Chayes ◽  
C. Winfield
Keyword(s):  
Correlation Length ◽  
First Passage Time ◽  
Passage Time ◽  
Scaling Behavior ◽  
First Passage Percolation ◽  
First Passage ◽  
Site Percolation ◽  
Percolation Problem ◽  
First Passage Problem

We introduce and study a novel type of first-passage percolation problem on where 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 probability p and (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 as p ↑ pc or p ↓ (1 – pc), the interface density vanishes with scaling behavior identical to the correlation length of the site percolation problem.

On first passage times of sticky reflecting diffusion processes with double exponential jumps

2020 ◽  
Vol 57 (1) ◽  
pp. 221-236 ◽  
Author(s):  
Shiyu Song ◽  
Yongjin Wang
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Explicit Solutions ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Double Exponential ◽  
Upper Level ◽  
First Passage Problem ◽  
Passage Times

AbstractWe explore the first passage problem for sticky reflecting diffusion processes with double exponential jumps. The joint Laplace transform of the first passage time to an upper level and the corresponding overshoot is studied. In particular, explicit solutions are presented when the diffusion part is driven by a drifted Brownian motion and by an Ornstein–Uhlenbeck process.

Moments of the first-passage time of a Wiener process with drift between two elastic barriers

1995 ◽  
Vol 32 (4) ◽  
pp. 1007-1013 ◽  
Author(s):  
Marco Dominé
Keyword(s):  
Wiener Process ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
One Dimensional ◽  
Special Cases ◽  
Backward Equation ◽  
The One ◽  
First Passage Problem ◽  
Reflecting Barriers

The first-passage problem for the one-dimensional Wiener process with drift in the presence of elastic boundaries is considered. We use the Kolmogorov backward equation with corresponding boundary conditions to derive explicit closed-form expressions for the expected value and the variance of the first-passage time. Special cases with pure absorbing and/or reflecting barriers arise for a certain choice of a parameter constellation.

First-passage percolation on the square lattice. I

1977 ◽  
Vol 9 (01) ◽  
pp. 38-54 ◽  
Author(s):  
R. T. Smythe ◽  
John C. Wierman
Keyword(s):  
Time Constant ◽  
First Passage Time ◽  
Passage Time ◽  
Square Lattice ◽  
Ergodic Theorems ◽  
First Passage Percolation ◽  
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.

Remarks on renewal theory for percolation processes

1976 ◽  
Vol 13 (02) ◽  
pp. 290-300 ◽  
Author(s):  
R. T. Smythe
Keyword(s):  
Ergodic Theorem ◽  
First Passage Time ◽  
Renewal Theory ◽  
Passage Time ◽  
Integer Lattice ◽  
Two Dimensional ◽  
First Passage Percolation ◽  
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.

First-passage percolation on the square lattice. I

1977 ◽  
Vol 9 (1) ◽  
pp. 38-54 ◽  
Author(s):  
R. T. Smythe ◽  
John C. Wierman
Keyword(s):  
Time Constant ◽  
First Passage Time ◽  
Passage Time ◽  
Square Lattice ◽  
Ergodic Theorems ◽  
First Passage Percolation ◽  
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.

scholarly journals Asymptotics for First-Passage Times on Delaunay Triangulations

2011 ◽  
Vol 20 (3) ◽  
pp. 435-453 ◽  
Author(s):  
LEANDRO P. R. PIMENTEL
Keyword(s):  
First Passage Time ◽  
Passage Time ◽  
Upper Bounds ◽  
Sufficient Condition ◽  
First Passage Percolation ◽  
First Passage ◽  
Delaunay Triangulations ◽  
Negative Constant ◽  
Additivity Theory ◽  
Passage Times

In this paper we study planar first-passage percolation (FPP) models on random Delaunay triangulations. In [14], Vahidi-Asl and Wierman showed, using sub-additivity theory, that the rescaled first-passage time converges to a finite and non-negative constant μ. We show a sufficient condition to ensure that μ>0 and derive some upper bounds for fluctuations. Our proofs are based on percolation ideas and on the method of martingales with bounded increments.

Remarks on renewal theory for percolation processes

1976 ◽  
Vol 13 (2) ◽  
pp. 290-300 ◽  
Author(s):  
R. T. Smythe
Keyword(s):  
Ergodic Theorem ◽  
First Passage Time ◽  
Renewal Theory ◽  
Passage Time ◽  
Integer Lattice ◽  
Two Dimensional ◽  
First Passage Percolation ◽  
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 L1-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.

A first-passage problem for a two-dimensional controlled random walk

1986 ◽  
Vol 23 (3) ◽  
pp. 670-678
Author(s):  
S. Lalley
Keyword(s):  
Random Walk ◽  
First Passage Time ◽  
Passage Time ◽  
Two Dimensions ◽  
Main Diagonal ◽  
Two Dimensional ◽  
First Passage ◽  
Explicit Representations ◽  
Mean Vectors ◽  
First Passage Problem

The process of interest is a controlled random walk in two dimensions: whenever the walker is above the main diagonal, the next increment to his position is chosen from a distribution FA; whenever the walker is below the diagonal, the next increment comes from another distribution FB. The two distributions have mean vectors which tend to push the walker back toward the diagonal. We analyze the problem of first passage to the first quadrant, obtaining explicit representations for the limiting first-entry distribution and expected first-passage time.

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