English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This work continues the investigation, initiated in a recent work byBenjamini and Sznitman, of percolative properties of the set of points notvisited by a random walk on the discrete torus (Z/NZ)^d up to time uN^d in highdimension d. If u>0 is chosen sufficiently small it has been shown that withoverwhelming probability this vacant set contains a unique giant componentcontaining segments of length c_0 log N for some constant c_0 > 0, and thiscomponent occupies a non-degenerate fraction of the total volume as N tends toinfinity. Within the same setup, we investigate here the complement of thegiant component in the vacant set and show that some components consist ofsegments of logarithmic size. In particular, this shows that the choice of asufficiently large constant c_0 > 0 is crucial in the definition of the giantcomponent.

Logarithmic components of the vacant set for random walk on a discrete torus