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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2603.29195 |
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| _version_ | 1866912990536663040 |
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| author | Wang, Qiao |
| author_facet | Wang, Qiao |
| contents | Let $F$ be the thermodynamic free energy of a ferromagnetic Ising model,analytic on $\mathbb{C}^{*}\setminus\mathcal{Z}_β$. The Lee--Yang edge at $z_c\in\partial\mathcal{Z}_β$ is characterised by $F(z)=F(z_c)+B(z-z_c)^{σ+1}+o(|z-z_c|^{σ+1})$ with $σ\in(-1,0)$ and $B\neq 0$.
We prove three results:
Theorem A (Jensen slope): defining the Jensen average $\widetilde{N}(x)=\frac{1}{2π}\int_0^{2π}\log|\widetilde{F}(e^{x+iθ})|\,dθ$ of $\widetilde{F}=F-F(z_c)$, the edge exponent satisfies $\widetilde{N}'(0^+)=σ+1$. The proof is a direct application of Jensen's formula.
Theorem B (Monodromy): the monodromy of $F$ around $z_c$ multiplies the singular part by $e^{2πi(σ+1)}$, a primitive $q$-th root of unity when $σ+1=p/q$.
Theorem C (Kac monodromy): for any 2D CFT at an RG fixed point with relevant operator $ϕ$ of weight $h_ϕ<0$ satisfying the Lee--Yang property, the RG scaling equation forces $σ=h_ϕ/(1-h_ϕ)$ and monodromy order $q=\mathrm{denom}(1/(1-h_ϕ))$.
We also prove that the edge expansion follows from the density asymptotics $ρ(θ)\sim A|θ-θ_c|^σ$ via a Mellin-transform calculation, making all three theorems unconditional for the $d=2$ Ising model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29195 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Lee--Yang Edge Exponent via Logarithmic Averaging Wang, Qiao Complex Variables Let $F$ be the thermodynamic free energy of a ferromagnetic Ising model,analytic on $\mathbb{C}^{*}\setminus\mathcal{Z}_β$. The Lee--Yang edge at $z_c\in\partial\mathcal{Z}_β$ is characterised by $F(z)=F(z_c)+B(z-z_c)^{σ+1}+o(|z-z_c|^{σ+1})$ with $σ\in(-1,0)$ and $B\neq 0$. We prove three results: Theorem A (Jensen slope): defining the Jensen average $\widetilde{N}(x)=\frac{1}{2π}\int_0^{2π}\log|\widetilde{F}(e^{x+iθ})|\,dθ$ of $\widetilde{F}=F-F(z_c)$, the edge exponent satisfies $\widetilde{N}'(0^+)=σ+1$. The proof is a direct application of Jensen's formula. Theorem B (Monodromy): the monodromy of $F$ around $z_c$ multiplies the singular part by $e^{2πi(σ+1)}$, a primitive $q$-th root of unity when $σ+1=p/q$. Theorem C (Kac monodromy): for any 2D CFT at an RG fixed point with relevant operator $ϕ$ of weight $h_ϕ<0$ satisfying the Lee--Yang property, the RG scaling equation forces $σ=h_ϕ/(1-h_ϕ)$ and monodromy order $q=\mathrm{denom}(1/(1-h_ϕ))$. We also prove that the edge expansion follows from the density asymptotics $ρ(θ)\sim A|θ-θ_c|^σ$ via a Mellin-transform calculation, making all three theorems unconditional for the $d=2$ Ising model. |
| title | The Lee--Yang Edge Exponent via Logarithmic Averaging |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2603.29195 |