Weak moment conditions for time coordinates in first-passage percolation models

1980 ◽  
Vol 17 (4) ◽  
pp. 968-978 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Time Constant ◽  
First Passage Times ◽  
First Passage Percolation ◽  
First Passage ◽  
Moment Conditions ◽  
Positive Order ◽  
Finite Moment ◽  
Time Coordinate ◽  
Point To Point ◽  
Passage Times

A generalization of first-passage percolation theory proves that the fundamental convergence theorems hold provided only that the time coordinate distribution has a finite moment of a positive order. The existence of a time constant is proved by considering first-passage times between intervals of sites, rather than the usual point-to-point and point-to-line first-passage times. The basic limit theorems for the related stochastic processes follow easily by previous techniques. The time constant is evaluated as 0 when the atom at 0 of the time-coordinate distribution exceeds½.

Weak moment conditions for time coordinates in first-passage percolation models

1980 ◽  
Vol 17 (04) ◽  
pp. 968-978 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Time Constant ◽  
First Passage Times ◽  
First Passage Percolation ◽  
First Passage ◽  
Moment Conditions ◽  
Positive Order ◽  
Finite Moment ◽  
Time Coordinate ◽  
Point To Point ◽  
Passage Times

A generalization of first-passage percolation theory proves that the fundamental convergence theorems hold provided only that the time coordinate distribution has a finite moment of a positive order. The existence of a time constant is proved by considering first-passage times between intervals of sites, rather than the usual point-to-point and point-to-line first-passage times. The basic limit theorems for the related stochastic processes follow easily by previous techniques. The time constant is evaluated as 0 when the atom at 0 of the time-coordinate distribution exceeds½.

First-passage percolation under weak moment conditions

1979 ◽  
Vol 16 (4) ◽  
pp. 750-763 ◽  
Author(s):  
Wolfgang Reh
Keyword(s):  
Time Constant ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Moment Conditions ◽  
Time Coordinate

Most research in first-passage percolation has been done under the assumption of a finite mean for the underlying time coordinate distribution. We demonstrate that the basic ergodic results can be derived under a weaker moment assumption, which still permits us to evaluate the time constant in the case where the atom at zero of the time coordinate distribution exceeds one-half. Further almost sure convergence is investigated more closely.

First-passage percolation under weak moment conditions

1979 ◽  
Vol 16 (04) ◽  
pp. 750-763 ◽  
Author(s):  
Wolfgang Reh
Keyword(s):  
Time Constant ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Moment Conditions ◽  
Time Coordinate

Most research in first-passage percolation has been done under the assumption of a finite mean for the underlying time coordinate distribution. We demonstrate that the basic ergodic results can be derived under a weaker moment assumption, which still permits us to evaluate the time constant in the case where the atom at zero of the time coordinate distribution exceeds one-half. Further almost sure convergence is investigated more closely.

First-passage percolation on the square lattice, II

1977 ◽  
Vol 9 (2) ◽  
pp. 283-295 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Sufficient Conditions ◽  
Initial Result ◽  
First Passage Times ◽  
Square Lattice ◽  
Maximum Height ◽  
First Passage Percolation ◽  
First Passage ◽  
Necessary And Sufficient ◽  
Passage Times

Several problems are considered in the theory of first-passage percolation on the two-dimensional integer lattice. The results include: (i) necessary and sufficient conditions for the existence of moments of first-passage times; (ii) determination of an upper bound for the time constant; (iii) an initial result concerning the maximum height of routes for first-passage times; (iv) ergodic theorems for a class of reach processes.

Total Positivity of Markov Chains and the Failure Rate Character of Some First Passage Times

1997 ◽  
Vol 29 (3) ◽  
pp. 713-732 ◽  
Author(s):  
Shiowjen Lee ◽  
J. Lynch
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Failure Rate ◽  
Total Positivity ◽  
First Passage Times ◽  
First Passage ◽  
Positive Order ◽  
Totally Positive ◽  
Ordered State ◽  
Passage Times

It is shown that totally positive order 2 (TP2) properties of the infinitesimal generator of a continuous-time Markov chain with totally ordered state space carry over to the chain's transition distribution function. For chains with such properties, failure rate characteristics of the first passage times are established. For Markov chains with partially ordered state space, it is shown that the first passage times have an IFR distribution under a multivariate total positivity condition on the transition function.

scholarly journals First passage percolation on sparse random graphs with boundary weights

2019 ◽  
Vol 56 (2) ◽  
pp. 458-471
Author(s):  
Lasse Leskelä ◽  
Hoa Ngo
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Time Scale ◽  
Network Size ◽  
First Passage Times ◽  
First Passage Percolation ◽  
First Passage ◽  
Connectivity Structure ◽  
Random Weights ◽  
Passage Times

AbstractA large and sparse random graph with independent exponentially distributed link weights can be used to model the propagation of messages or diseases in a network with an unknown connectivity structure. In this article we study an extended setting where, in addition, the nodes of the graph are equipped with nonnegative random weights which are used to model the effect of boundary delays across paths in the network. Our main results provide approximative formulas for typical first passage times, typical flooding times, and maximum flooding times in the extended setting, over a time scale logarithmic with respect to the network size.

A Note on a Problem by Welsh in First-Passage Percolation

1998 ◽  
Vol 7 (1) ◽  
pp. 11-15 ◽  
Author(s):  
SVEN ERICK ALM
Keyword(s):  
First Passage Times ◽  
Square Lattice ◽  
Trivial Case ◽  
First Passage Percolation ◽  
First Passage ◽  
General Proof ◽  
Mean First Passage Times ◽  
The Subject ◽  
Passage Times

Consider first-passage percolation on the square lattice. Welsh, who together with Hammersley introduced the subject in 1963, has formulated a problem about mean first-passage times, which, although seemingly simple, has not been proved in any non-trivial case. In this paper we give a general proof of Welsh's problem.

First-passage percolation on the square lattice, II

1977 ◽  
Vol 9 (02) ◽  
pp. 283-295 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Sufficient Conditions ◽  
Initial Result ◽  
First Passage Times ◽  
Square Lattice ◽  
Maximum Height ◽  
First Passage Percolation ◽  
First Passage ◽  
Necessary And Sufficient ◽  
Passage Times

Several problems are considered in the theory of first-passage percolation on the two-dimensional integer lattice. The results include: (i) necessary and sufficient conditions for the existence of moments of first-passage times; (ii) determination of an upper bound for the time constant; (iii) an initial result concerning the maximum height of routes for first-passage times; (iv) ergodic theorems for a class of reach processes.

Total Positivity of Markov Chains and the Failure Rate Character of Some First Passage Times

1997 ◽  
Vol 29 (03) ◽  
pp. 713-732 ◽  
Author(s):  
Shiowjen Lee ◽  
J. Lynch
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Failure Rate ◽  
Total Positivity ◽  
First Passage Times ◽  
First Passage ◽  
Positive Order ◽  
Totally Positive ◽  
Ordered State ◽  
Passage Times

It is shown that totally positive order 2 (TP2) properties of the infinitesimal generator of a continuous-time Markov chain with totally ordered state space carry over to the chain's transition distribution function. For chains with such properties, failure rate characteristics of the first passage times are established. For Markov chains with partially ordered state space, it is shown that the first passage times have an IFR distribution under a multivariate total positivity condition on the transition function.

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