2024/12/12 by Biskup, Marek, Huang, Haiyu
#82B26 #82B27 #82B28 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2412.08964
Given a square box Λn⊆\mathbb Z2 of side length Ln with L,n>1, we study hierarchical random fields \ϕx\colon x∈Λn\ with law proportional to \rm e\frac12β(ϕ,Δnϕ)∏x∈Λnν(\rm dϕx), where β>0 is the inverse temperature, Δn is a hierarchical Laplacian on Λn, and ν is a non-degenerate 1-periodic measure on \mathbb R. Our setting includes the integer-valued Gaussian field (a.k.a. DG model or Villain Coulomb gas) and the sine-Gordon model. Relying on renormalization group analysis we derive sharp asymptotic formulas, in the limit as n→∞, for the covariance ⟨ϕxϕy⟩ and the fractional charge ⟨ \rm e^2π\rm iα(ϕx-ϕy)⟩ in the subcritical β<β\rm c:=π2/log L, critical β=β\rm c and slightly supercritical β>β\rm c regimes. The field exhibits logarithmic correlations throughout albeit with a distinct β-dependence of both the covariance scale and the fractional-charge exponents in the sub/supercritical regimes. Explicit logarithmic corrections appear at the critical point.