Analyticity for locally stable hard-core gases via recursion
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916876260474880 |
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| author | He, Qidong |
| author_facet | He, Qidong |
| contents | In their recent works [Comm. Math. Phys. 399:1 (2023)] and [arXiv:2109.01094], Michelen and Perkins proved that the pressure of a system of particles with repulsive pair interactions is analytic for activities up to $eΔ_ϕ(β)^{-1}$, where $Δ_ϕ(β)\in(0,C_ϕ(β)]$ is a constant they called the potential-weighted connective constant. This paper extends their method to locally stable, tempered, and hard-core pair potentials. Our main result is that the pressure of such a system is analytic for activities up to $e^{2-2W(eA_ϕ(β)/Δ_ϕ(β))}Δ_ϕ(β)^{-1}e^{-(βC+1)}$, where $C\ge0$ is the local stability constant, $W(\cdot)$ the Lambert $W$-function, $A_ϕ(β)$ the contribution from the attraction in the pair potential to the temperedness constant, and $Δ_ϕ(β)\in[A_ϕ(β),C_ϕ(β)]$ a counterpart of the constant defined by Michelen and Perkins. The main ingredients in the proof include a recursive identity for the one-point density tailored to locally stable hard-core potentials and a corresponding notion of modulations of an activity function. In the high-temperature regime, our result surpasses the classical Penrose-Ruelle bound of $C_ϕ(β)^{-1}e^{-(βC+1)}$ by at least a factor of $e^{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04451 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Analyticity for locally stable hard-core gases via recursion He, Qidong Mathematical Physics Statistical Mechanics Probability In their recent works [Comm. Math. Phys. 399:1 (2023)] and [arXiv:2109.01094], Michelen and Perkins proved that the pressure of a system of particles with repulsive pair interactions is analytic for activities up to $eΔ_ϕ(β)^{-1}$, where $Δ_ϕ(β)\in(0,C_ϕ(β)]$ is a constant they called the potential-weighted connective constant. This paper extends their method to locally stable, tempered, and hard-core pair potentials. Our main result is that the pressure of such a system is analytic for activities up to $e^{2-2W(eA_ϕ(β)/Δ_ϕ(β))}Δ_ϕ(β)^{-1}e^{-(βC+1)}$, where $C\ge0$ is the local stability constant, $W(\cdot)$ the Lambert $W$-function, $A_ϕ(β)$ the contribution from the attraction in the pair potential to the temperedness constant, and $Δ_ϕ(β)\in[A_ϕ(β),C_ϕ(β)]$ a counterpart of the constant defined by Michelen and Perkins. The main ingredients in the proof include a recursive identity for the one-point density tailored to locally stable hard-core potentials and a corresponding notion of modulations of an activity function. In the high-temperature regime, our result surpasses the classical Penrose-Ruelle bound of $C_ϕ(β)^{-1}e^{-(βC+1)}$ by at least a factor of $e^{2}$. |
| title | Analyticity for locally stable hard-core gases via recursion |
| topic | Mathematical Physics Statistical Mechanics Probability |
| url | https://arxiv.org/abs/2405.04451 |