2019/12/22 by Giambattista Giacomin, Giacomin, Giambattista, Hubert Lacoin +1
Materials Science · Mathematics · Physics and Astronomy · #Block Copolymer Self-Assembly #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1912.10538
openalex publication_date 2019/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We continue the study, initiated in [Giacomin and Lacoin, JEMS 2018], of the localization transition of a lattice free field ϕ=(ϕ(x))x ∈ Zd, d≥ 3, in presence of a quenched disordered substrate. The presence of the substrate affects the interface at the spatial sites in which the interface height is close to zero. This corresponds to the Hamiltonian ∑x∈ Zd (βωx+h)δx, where δx=1[-1,1](ϕ(x)), and (ωx)x∈ Zd is an IID centered field. A transition takes place when the average pinning potential h goes past a threshold hc(β): from a delocalized phase hhc(β) where the field sticks to the substrate. In [Giacomin and Lacoin, JEMS 2018] the critical value of h is identified and it coincides, up to the sign, with the log-Laplace transform of ω=ωx, that is -hc(β)=λ(β):=log E[eβω]. Here we obtain the sharp critical behavior of the free energy approaching criticality: limu\searrow 0 ( F(β,hc(β)+u))/(u2)= \frac12 \textrmVar(eβω-λ(β)). Moreover, we give a precise description of the trajectories of the field in the same regime: the absolute value of the field is √(2σd2\vertlog(h-hc(β))\vert) to leading order when h\searrow hc(β) except on a vanishing fraction of sites (σd2 is the single site variance of the free field).