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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2512.04394 |
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| _version_ | 1866910076462170112 |
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| author | Cuenca, Cesar Xu, Jiaming |
| author_facet | Cuenca, Cesar Xu, Jiaming |
| contents | In this paper, we find necessary and sufficient conditions for the Law of Large Numbers of averaged empirical measures of $N$-particle ensembles, in terms of the asymptotics of their Bessel generating functions, in the fixed temperature regime. This settles an open problem posed by Benaych-Georges, Cuenca and Gorin. For one direction, we use the moment method through Dunkl operators, and for the other we employ a special case of the formula of Chapuy--Dolega for the generating function of infinite constellations. As applications, we prove that the LLN for $θ$-sums and $θ$-corners of random matrices are given by the free convolution and free projection, respectively, regardless of the value of inverse temperature parameter $θ$. We also prove the LLN for a time-slice of the $θ$-Dyson Brownian motion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04394 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Law of Large Numbers for continuous $N$-particle ensembles at fixed temperature Cuenca, Cesar Xu, Jiaming Probability Mathematical Physics Combinatorics In this paper, we find necessary and sufficient conditions for the Law of Large Numbers of averaged empirical measures of $N$-particle ensembles, in terms of the asymptotics of their Bessel generating functions, in the fixed temperature regime. This settles an open problem posed by Benaych-Georges, Cuenca and Gorin. For one direction, we use the moment method through Dunkl operators, and for the other we employ a special case of the formula of Chapuy--Dolega for the generating function of infinite constellations. As applications, we prove that the LLN for $θ$-sums and $θ$-corners of random matrices are given by the free convolution and free projection, respectively, regardless of the value of inverse temperature parameter $θ$. We also prove the LLN for a time-slice of the $θ$-Dyson Brownian motion. |
| title | Law of Large Numbers for continuous $N$-particle ensembles at fixed temperature |
| topic | Probability Mathematical Physics Combinatorics |
| url | https://arxiv.org/abs/2512.04394 |