A phase transition in the Bakry-Émery gradient estimate for Dyson Brownian motion
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915327238995968 |
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| author | Suzuki, Kohei Tashiro, Kenshiro |
| author_facet | Suzuki, Kohei Tashiro, Kenshiro |
| contents | In this paper, we find a gap between the lower bound of the Bakry-Émery $N$-Ricci tensor ${\rm Ric}_N$ and the Bakry-Émery gradient estimate ${\sf BE}$ in the space associated with the finite-particle Dyson Brownian motion (DBM) with inverse temperature $0<β<1$. Namely, we prove that, for the weighted space $(\mathbb R^n, w_β)$ with $w_β=\prod_{i<j}^n |x_i-x_j|^β$ and any $N\in[n+\fracβ{2}n(n-1),+\infty]$,
$β\ge 1 \implies {\rm Ric}_N \ge 0 \ \& \ {\sf BE}(0,N)$ hold;
$0 < β< 1 \implies {\rm Ric}_N \ge 0$ holds while ${\sf BE}(0,N)$ does not, which shows a phase transition of the Dyson Brownian motion regarding the Bakry-Émery curvature bound in the small inverse temperature regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04424 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A phase transition in the Bakry-Émery gradient estimate for Dyson Brownian motion Suzuki, Kohei Tashiro, Kenshiro Probability Mathematical Physics Functional Analysis In this paper, we find a gap between the lower bound of the Bakry-Émery $N$-Ricci tensor ${\rm Ric}_N$ and the Bakry-Émery gradient estimate ${\sf BE}$ in the space associated with the finite-particle Dyson Brownian motion (DBM) with inverse temperature $0<β<1$. Namely, we prove that, for the weighted space $(\mathbb R^n, w_β)$ with $w_β=\prod_{i<j}^n |x_i-x_j|^β$ and any $N\in[n+\fracβ{2}n(n-1),+\infty]$, $β\ge 1 \implies {\rm Ric}_N \ge 0 \ \& \ {\sf BE}(0,N)$ hold; $0 < β< 1 \implies {\rm Ric}_N \ge 0$ holds while ${\sf BE}(0,N)$ does not, which shows a phase transition of the Dyson Brownian motion regarding the Bakry-Émery curvature bound in the small inverse temperature regime. |
| title | A phase transition in the Bakry-Émery gradient estimate for Dyson Brownian motion |
| topic | Probability Mathematical Physics Functional Analysis |
| url | https://arxiv.org/abs/2506.04424 |