Mixing-time fluctuations for random walk on a random-walk range

Determine the second-order fluctuations and limiting distribution of the mixing time for a random walk run on the graph whose vertices are the points of a simple random-walk range {S_0,...,S_n}.

Background

The paper’s long-range-intersection analysis is proposed as a tool for studying dynamical quantities of a random walk performed on the random-walk range. Long-range intersections determine shortcuts in the underlying graph and are therefore expected to influence the mixing behavior.

The authors leave unresolved both the second-order fluctuation scale and the limiting distribution of the mixing time for this process.

References

The structure of the long range intersections would also be key to deriving mixing time estimates for a random walk run on the random walk range. Consider a random walk run on the graph of the random walk range ${S_0,\ldots,S_n}$. Determine the second-order fluctuations and limiting distribution of the mixing time of this process.

— Graph distance and effective resistance of the random walk trace in four and five dimensions  (2608.23135 - Adhikari et al., 24 Aug 2026) in Section 1, subsection “Open Problems”