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PreprintCC BY 4.0arXiv

Entropy, isoperimetry and the total variation profile of random walks

  • Antoine Gournay
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Abstract

The total variation profile TVn of a random walk measures how far its distribution at time n is from being invariant under the generators. Three lower bounds are given: an elementary bound TVn ≥ 1/(44 L(n)) in terms of the speed L(n); a bound by the entropy increment H(Pn+1) − H(Pn), which is a reverse Pinsker inequality; and a bound in terms of the entropy and the isoperimetric profile. Deciding whether the entropy-increment bound always improves on the isoperimetric one is related to conjectures on the growth of the entropy and of groups. The motivation comes from the question of whether the distributions of a random walk can be transported back onto their translates at bounded cost.

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Cite

BibTeXRIS
BibTeX
@misc{gournay2026entropy,
  title = {{Entropy, isoperimetry and the total variation profile of random walks}},
  author = {Antoine Gournay},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16834},
  eprint = {2610.16834},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16834},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Entropy, isoperimetry and the total variation profile of random walks
AU  - Antoine Gournay
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16834
UR  - https://arxiv.org/abs/2610.16834
AB  - The total variation profile TV_n of a random walk measures how far its distribution at time n is from being invariant under the generators. Three lower bounds are given: an elementary bound TV_n ≥ 1/(44 L(n)) in terms of the speed L(n); a bound by the entropy increment H(P^n+1) − H(P^n), which is a reverse Pinsker inequality; and a bound in terms of the entropy and the isoperimetric profile. Deciding whether the entropy-increment bound always improves on the isoperimetric one is related to conjectures on the growth of the entropy and of groups. The motivation comes from the question of whether the distributions of a random walk can be transported back onto their translates at bounded cost.
ER  -