Small entropy doubling for random walks and polynomial growth
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908949760966656 |
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| author | Blachar, Guy |
| author_facet | Blachar, Guy |
| contents | Gromov's theorem states that a finitely generated group has polynomial growth if and only if it is virtually nilpotent. A key ingredient in its proof is the small doubling property. In this work, we study entropy analogues of this property for random walks on groups. We show that if a finitely supported symmetric random walk $R_n$ satisfies \[ \mathrm{H}(R_{2n}) \le \mathrm{H}(R_n) + \log K \] at some sufficiently large scale $n$, then the underlying group is virtually nilpotent, with bounds depending on $K$ and $μ_{\min}$.
Our approach adapts Tao's entropy Balog--Szemerédi--Gowers argument to unimodular locally compact groups, combined with structural results on approximate groups.
As applications, we obtain entropy-based criteria for polynomial growth. We also deduce an entropy gap phenomenon: if $G$ is not virtually nilpotent, then the entropy of random walks on $G$ grows faster than a universal superlogarithmic function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_08490 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Small entropy doubling for random walks and polynomial growth Blachar, Guy Group Theory Probability Gromov's theorem states that a finitely generated group has polynomial growth if and only if it is virtually nilpotent. A key ingredient in its proof is the small doubling property. In this work, we study entropy analogues of this property for random walks on groups. We show that if a finitely supported symmetric random walk $R_n$ satisfies \[ \mathrm{H}(R_{2n}) \le \mathrm{H}(R_n) + \log K \] at some sufficiently large scale $n$, then the underlying group is virtually nilpotent, with bounds depending on $K$ and $μ_{\min}$. Our approach adapts Tao's entropy Balog--Szemerédi--Gowers argument to unimodular locally compact groups, combined with structural results on approximate groups. As applications, we obtain entropy-based criteria for polynomial growth. We also deduce an entropy gap phenomenon: if $G$ is not virtually nilpotent, then the entropy of random walks on $G$ grows faster than a universal superlogarithmic function. |
| title | Small entropy doubling for random walks and polynomial growth |
| topic | Group Theory Probability |
| url | https://arxiv.org/abs/2604.08490 |