# Some problems concerning the structure of random walk paths

@article{Erdos1963SomePC, title={Some problems concerning the structure of random walk paths}, author={Paul L. Erdos and Stephen Taylor}, journal={Acta Mathematica Academiae Scientiarum Hungarica}, year={1963}, volume={11}, pages={137-162} }

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 are integers, and let a point S,~(n) perform a move randomly on this lattice according to the rules: at time zero it is at the origin and if at any time n-1 ( n ~ l, 2, . ..) it is at some point S of the lattice, then at time n it will be at one of the 2 d lattice points nearest S, the probability…

## 172 Citations

A stable limit law for recurrence times of the simple random walk on the two-dimensional integer lattice

- Mathematics
- 2014

We consider the random walk of a particle on the two-dimensional integer lattice starting at the origin and moving from each site (independently of the previous moves) with equal probabilities to any…

Random Walks on Lattices. II

- Mathematics
- 1965

Formulas are obtained for the mean first passage times (as well as their dispersion) in random walks from the origin to an arbitrary lattice point on a periodic space lattice with periodic boundary…

Brownian Intersections, Cover Times and Thick Points via Trees

- Mathematics
- 2002

There is a close connection between intersections of Brownian motion paths and percolation on trees. Recently, ideas from probability on trees were an important component of the multifractal analysis…

On multiple visits in lattice random walks

- Mathematics
- 1988

In the present work we treat in detail the problem of multiple visits in lattice random walks. We show that this problem is closely related to the well-studied property of the number of distinct…

On Escaping, Entering, and Visiting Discs of Projections of Planar Symmetric Random Walks on the Lattice Torus

- Mathematics
- 2012

We examine escape and entrance times, Green's functions, local times, and hitting distributions of discs and annuli of a symmetric random walk on $\Z^2$ projected onto the periodic lattice $\Z^2_K$.…

Thick points of random walk and the Gaussian free field

- MathematicsElectronic Journal of Probability
- 2020

We consider the thick points of random walk, i.e. points where the local time is a fraction of the maximum. In two dimensions, we answer a question of Dembo, Peres, Rosen and Zeitouni and compute the…

Late Points and Cover Times of Projections of Planar Symmetric Random Walks on the Lattice Torus

- Mathematics
- 2014

We examine the sets of late points of a symmetric random walk on $Z^2$ projected onto the torus $Z^2_K$, culminating in a limit theorem for the cover time of the toral random walk. This extends the…

Intersections of random walks

- Mathematics
- 2011

We study the large deviation behaviour of simple random walks in dimension three or more in this thesis. The first part of the thesis concerns the number of lattice sites visited by the random walk.…

## References

SHOWING 1-9 OF 9 REFERENCES

Some Problems on Random Walk in Space

- Mathematics
- 1951

Consider the lattice formed by all points whose coordinates are integers in d-dimensional Euclidean space, and let a point S j(n) perform a move randomly on this lattice according to the following…

Some intersection properties of random walk paths

- Mathematics
- 1960

We complete the solution of this problem in Section 3 . Clearly, there is no problem for d-== I or 2 . The solution takes a different form in the cases d= 3, d-4, and d = 5. For example, if d = 4, an…

On the random walk and Brownian motion

- Mathematics
- 1962

(Wiener process) X(t), 0 t < oo, with X(0) = 0, and on the other a classical random walk S(n) = ,'J= Xi, 1 _ n < co, where X1, X2, * * * is a sequence of Bernoulli trials with probability 1/2 for Xi=…

Random Walk and the Theory of Brownian Motion

- Mathematics
- 1947

(1947). Random Walk and the Theory of Brownian Motion. The American Mathematical Monthly: Vol. 54, No. 7P1, pp. 369-391.

Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz

- Mathematics
- 1921

On the zeros of + 1

- Annals of Math
- 1949

Brownian motion in space and subharmonic functions (under press)

- Brownian motion in space and subharmonic functions (under press)