# Recurrent Graphs where Two Independent Random Walks Collide Finitely Often

@article{Krishnapur2004RecurrentGW, title={Recurrent Graphs where Two Independent Random Walks Collide Finitely Often}, author={Manjunath Krishnapur and Yuval Peres}, journal={Electronic Communications in Probability}, year={2004}, volume={9}, pages={72-81} }

We present a class of graphs where simple random walk is recurrent, yet two independent walkers meet only finitely many times almost surely. In particular, the comb lattice, obtained from $Z^2$ by removing all horizontal edges off the $x$-axis, has this property. We also conjecture that the same property holds for some other graphs, including the incipient infinite cluster for critical percolation in $Z^2$.

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## References

SHOWING 1-10 OF 12 REFERENCES

### Random Walks on Infinite Graphs and Groups

- Mathematics
- 2000

Part I. The Type Problem: 1. Basic facts 2. Recurrence and transience of infinite networks 3. Applications to random walks 4. Isoperimetric inequalities 5. Transient subtrees, and the classification…

### Some problems concerning the structure of random walk paths

- Mathematics
- 1963

1. In t roduct ion . We restrict our consideration to symmetric random walk, defined in the following way. Consider the lattice formed by the points of d-dimensional Euclidean space whose coordinates…

### The incipient infinite cluster in two-dimensional percolation

- Mathematics
- 1986

SummaryLetPp be the probability measure on the configurations of occupied and vacant vertices of a two-dimensional graphG, under which all vertices are independently occupied (respectively vacant)…

### A characterization of the invariant measures for an infinite particle system with interactions

- Mathematics
- 1973

ABSTRACT. Let p(x,y) be the transition function for a symmetric, irreducible, transient Markov chain on the countable set S. Let 71 be the infinite particle system on S with the simple exclusion…

### Subdiffusive behavior of random walk on a random cluster

- Mathematics
- 1986

On considere deux cas de marche aleatoire {X n } n≥0 sur un graphe aleatoire #7B-G. Dans le cas ou #7B-G est l'arbre d'un processus de branchement critique, conditionne par la non-extinction, si h(x)…

### An Introduction to Probability Theory and Its Applications

- Mathematics
- 1950

Thank you for reading an introduction to probability theory and its applications vol 2. As you may know, people have look numerous times for their favorite novels like this an introduction to…

### An Introduction To Probability Theory And Its Applications

- Mathematics
- 1950

A First Course in Probability (8th ed.) by S. Ross is a lively text that covers the basic ideas of probability theory including those needed in statistics.

### Probability Trees

- MathematicsGraphics Interface
- 1997

A k D tree representation of probability distribu tions is generalized to support generation of samples from conditional distributions An interpretation of the approach as a piecewise linear warping…