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.

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.

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.

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.

On ageing properties of first-passage times of increasing Markov processes

2002 ◽  
Vol 34 (01) ◽  
pp. 241-259
Author(s):  
Félix Belzunce ◽  
Eva-María Ortega ◽  
José M. Ruiz
Keyword(s):  
Markov Chain ◽  
Markov Processes ◽  
Continuous Time ◽  
Transition Matrix ◽  
Sufficient Conditions ◽  
First Passage Times ◽  
Generator Matrix ◽  
First Passage ◽  
Finite Case ◽  
Passage Times

The purpose of this paper is to study ageing properties of first-passage times of increasing Markov chains. We extend the literature to some new ageing classes, such as the IFR(2), NBU(2), DRLLt and NBULt classes. We also give sufficient conditions in the finite case, that are more efficient computationally, just in terms of the transition matrix K, in the discrete case, or the generator matrix Q, in the continuous case. For the uniformizable, continuous-time Markov processes, we derive conditions in terms of the discrete uniformized Markov chain for the NBU(2) and the NBULt classes. In the last section, a review of the main results in this direction in the literature is given, and we compare some of the conditions stated in this paper with others given in the literature about some other ageing classes. Some examples where these results are applied are given.

On ageing properties of first-passage times of increasing Markov processes

2002 ◽  
Vol 34 (1) ◽  
pp. 241-259 ◽  
Author(s):  
Félix Belzunce ◽  
Eva-María Ortega ◽  
José M. Ruiz
Keyword(s):  
Markov Chain ◽  
Markov Processes ◽  
Continuous Time ◽  
Transition Matrix ◽  
Sufficient Conditions ◽  
First Passage Times ◽  
Generator Matrix ◽  
First Passage ◽  
Finite Case ◽  
Passage Times

The purpose of this paper is to study ageing properties of first-passage times of increasing Markov chains. We extend the literature to some new ageing classes, such as the IFR(2), NBU(2), DRLLt and NBULt classes. We also give sufficient conditions in the finite case, that are more efficient computationally, just in terms of the transition matrix K, in the discrete case, or the generator matrix Q, in the continuous case. For the uniformizable, continuous-time Markov processes, we derive conditions in terms of the discrete uniformized Markov chain for the NBU(2) and the NBULt classes. In the last section, a review of the main results in this direction in the literature is given, and we compare some of the conditions stated in this paper with others given in the literature about some other ageing classes. Some examples where these results are applied are given.

First passage times and lumpability of semi-Markov processes

1988 ◽  
Vol 25 (04) ◽  
pp. 675-687 ◽  
Author(s):  
Ushio Sumita ◽  
Maria Rieders
Keyword(s):  
Markov Processes ◽  
First Passage Times ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
First Passage ◽  
The Laplace Transform ◽  
First Exit Times ◽  
Semi Markov Processes ◽  
Necessary And Sufficient ◽  
Passage Times

A necessary and sufficient condition of Serfozo (1971) for lumpability of semi-Markov processes is reinterpreted in terms of first-exit times. Furthermore, a new necessary and sufficient condition is developed by establishing relationships between first-passage times and lumpability of semi-Markov processes. The approach taken in this paper is entirely based on the Laplace-transform domain.

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.

On the first-passage times of pure jump processes

1988 ◽  
Vol 25 (3) ◽  
pp. 501-509 ◽  
Author(s):  
Moshe Shaked ◽  
J. George Shanthikumar
Keyword(s):  
Failure Rate ◽  
Sufficient Conditions ◽  
Jump Process ◽  
First Passage Times ◽  
Jump Processes ◽  
First Passage ◽  
Increasing Failure Rate ◽  
Passage Times

Let Tx be the time it takes for a pure jump process, which starts at 0, to cross a threshold x > 0. Sufficient conditions on the parameters of this process under which Tx has increasing failure rate average (IFRA), increasing failure rate (IFR) or logconcave density (PF2) are identified. The conditions for IFRA are weaker than those of Drosen (1986). Sufficient conditions on the parameter of a pure jump process for Tx to the IFR or PF2 are not available in the literature.

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