# Multiple Random Walks and Interacting Particle Systems

## Abstract

We study properties of multiple random walks on a graph under various assumptions of interaction between the particles. To give precise results, we make our analysis for random regular graphs. The cover time of a random walk on a random *r*-regular graph was studied in [6], where it was shown with high probability (**whp**), that for *r* ≥ 3 the cover time is asymptotic to *θ* _{ r } *n* ln *n*, where *θ* _{ r } = (*r* − 1)/(*r* − 2). In this paper we prove the following (**whp**) results. For *k* independent walks on a random regular graph *G*, the cover time *C* _{ G }(*k*) is asymptotic to *C* _{ G }/*k*, where *C* _{ G } is the cover time of a single walk. For most starting positions, the expected number of steps before any of the walks meet is \(\theta_r n/\binom{k}{2}\). If the walks can communicate when meeting at a vertex, we show that, for most starting positions, the expected time for *k* walks to broadcast a single piece of information to each other is asymptotic to 2*θ* _{ r } *n* (ln *k*)/*k*, as *k*,*n* → ∞.

We also establish properties of walks where there are two types of particles, predator and prey, or where particles interact when they meet at a vertex by coalescing, or by annihilating each other. For example, the expected extinction time of *k* explosive particles (*k* even) tends to (2ln 2) *θ* _{ r } *n* as *k*→ ∞.

The case of *n* coalescing particles, where one particle is initially located at each vertex, corresponds to a voter model defined as follows: Initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbour. The expected time for a unique opinion to emerge is the expected time for all the particles to coalesce, which is asymptotic to 2 *θ* _{ r } *n*.

Combining results from the predator-prey and multiple random walk models allows us to compare expected detection time in the following cops and robbers scenarios: both the predator and the prey move randomly, the prey moves randomly and the predators stay fixed, the predators move randomly and the prey stays fixed. In all cases, with *k* predators and ℓ prey the expected detection time is *θ* _{ r } *H* _{ℓ} *n*/*k*, where *H* _{ℓ} is the ℓ-th harmonic number.

## Keywords

Random Walk Voter Model Cover Time Interact Particle System Small Positive Constant## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Aldous, D.: Some inequalities for reversible Markov chains. J. London Math. Soc. 25(2), 564–576 (1982)MathSciNetCrossRefMATHGoogle Scholar
- 2.Aldous, D., Fill, J.: Reversible Markov Chains and Random Walks on Graphs. Monograph in preparation, http://stat-www.berkeley.edu/pub/users/aldous/RWG/book.html
- 3.Aleliunas, R., Karp, R.M., Lipton, R.J., Lovász, L., Rackoff, C.: Random Walks, Universal Traversal Sequences, and the Complexity of Maze Problems. In: Proceedings of the 20th Annual IEEE Symposium on Foundations of Computer Science, pp. 218–223 (1979)Google Scholar
- 4.Alon, N., Avin, C., Kouchý, M., Kozma, G., Lotker, Z., Tuttle, M.: Many random walks are faster then one. In: Proceedings of the 20th Annual ACM Symposium on Parallelism in Algorithms and Architectures, pp. 119–128 (2008)Google Scholar
- 5.Broder, A., Karlin, A., Raghavan, A., Upfal, E.: Trading space for time in undirected s-t connectivity. In: Proceedings of the 21st Annual ACM Symposium on Theory of Computing, pp. 543–549 (1989)Google Scholar
- 6.Cooper, C., Frieze, A.M.: The cover time of random regular graphs. SIAM Journal on Discrete Mathematics 18, 728–740 (2005)MathSciNetCrossRefMATHGoogle Scholar
- 7.Cooper, C., Frieze, A.M.: The cover time of the preferential attachment graph. Journal of Combinatorial Theory Series B 97(2), 269–290 (2007)MathSciNetCrossRefMATHGoogle Scholar
- 8.Cooper, C., Frieze, A.M.: The cover time of the giant component of a random graph. Random Structures and Algorithms 32, 401–439 (2008)MathSciNetCrossRefMATHGoogle Scholar
- 9.Cooper, C., Frieze, A.M., Radzik, T.: Multiple random walks in random regular graphs (2008), http://www.math.cmu.edu/~af1p/Texfiles/Multiple.pdf
- 10.Feige, U.: A tight upper bound for the cover time of random walks on graphs. Random Structures and Algorithms 6, 51–54 (1995)MathSciNetCrossRefMATHGoogle Scholar
- 11.Feige, U.: A tight lower bound for the cover time of random walks on graphs. Random Structures and Algorithms 6, 433–438 (1995)MathSciNetCrossRefMATHGoogle Scholar
- 12.Feller, W.: An Introduction to Probability Theory, 2nd edn., vol. I. Wiley, Chichester (1960)Google Scholar
- 13.Friedman, J.: A proof of Alon’s second eignevalue conjecture. Memoirs of the A.M.S. (to appear)Google Scholar
- 14.Liggett, T.M.: Interacting Particle Systems. Springer, Heidelberg (1985)CrossRefMATHGoogle Scholar
- 15.Lovász, L.: Random walks on graphs: a survey. In: Bolyai Society Mathematical Studies, Combinatorics, Paul Erdös is Eighty, Keszthely, Hungary, vol. 2, pp. 1–46 (1993)Google Scholar
- 16.Sinclair, A.: Improved bounds for mixing rates of Markov chains and multicommodity flow. Combinatorics, Probability and Computing 1, 351–370 (1992)MathSciNetCrossRefMATHGoogle Scholar