Analyticity for Locally Stable Hard-Core Gases via Recursion

Abstract In their recent works [Comm Math Phys 399:367–388 (2023)] and [Comm Math Phys 406:32 (2025)], Michelen and Perkins proved that the pressure of a system of particles with repulsive pair interactions is analytic for activities in a complex neighborhood of $$[0,e\Delta _{\phi }(\beta )^{-1})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>e</mml:mi> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>β</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where $$\Delta _{\phi }(\beta )\in (0,C_{\phi }(\beta )]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>β</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>β</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> denotes what they call the potential-weighted connective constant. This paper extends their method to locally stable (possibly attractive), tempered, and hard-core pair potentials. We obtain an analogous analyticity result that is most effective in the high-temperature regime, where it surpasses the classical Penrose-Ruelle bound of $$C_{\phi }(\beta )^{-1}e^{-(\beta C+1)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>β</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mo>(</mml:mo> <mml:mi>β</mml:mi> <mml:mi>C</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> by at least a factor of $$e^{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>e</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> . 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.