Existence of transient two-dimensional random walks on discrete point processes

About 16 years old · traced to 2010

A two-dimensional random walk on a discrete point process is a random walk in dimension $2$ whose state space is a discrete point process. Existence conjecture. There are transient two-dimensional random walks on discrete point processes. The paper notes that transience and recurrence are fully classified in dimensions $d=1$ and $d\geq 3$ , but only partially classified in dimension $2$ ; this conjecture asserts that transience can occur there.

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Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Existence of transient two-dimensional random walks on discrete point processes
A two-dimensional random walk on a discrete point process is the random walk considered in the paper when the ambient dimension is $2$ . Existence conjecture. There are transient two-dimensional random walks on discrete point processes. The paper states this as an open direction because recurrence and transience are fully classified in dimensions $1$ and at least $3$ , but only partially classified in dimension $2$ .
source: Ron Rosenthal, “Random walk on discrete point processes”, arXiv:1005.1398 (2011).

References

Primary source

Noam Berger and Ron Rosenthal, “Behavior of random walk on discrete point processes”, arXiv:1110.5740 (2013).

Additional references

2 papers in this index state this conjecture (2010–2011). The statement above is taken from the most recent of them; the others are arXiv:1005.1398.

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