Kamil Khettabi | Searcharxiv

arXiv2026

Typical geometry of self-repelling polymers in a constant force field

We study a general class of self-repelling polymers on $\mathbb Z^2$, including the simple random walk, the self-avoiding walk and the repulsive Domb-Joyce model, in the presence of a constant force field acting on each monomer. Conditioning the polymer to have fixed length and fixed endpoints, we identify the limiting free energy and prove that typical trajectories concentrate exponentially near a deterministic macroscopic shape. This shape is characterized as the unique minimizer of a variational problem and can be interpreted as a geodesic of a height-dependent Finsler metric. We also analyze two limiting regimes with universal features: for small field strength, in the symmetric case, the geodesic is close to a classical catenary, while for large field strength it converges to a universal polygonal shape governed by the nearest-neighbor lattice constraint.

math-ph↗

arXiv2023

On the two-point function of the Ising model with infinite range-interactions

In this article, we prove some results concerning the truncated two-point function of the infinite-range Ising model above and below the critical temperature. More precisely, if the coupling constants are of the form $J_{x}= ψ(x)e^{ -ρ(x)}$ with $ρ$ some norm and $ψ$ an subexponential correction, we show under appropriate assumptions that given $s\in\mathbb{S}^{d-1}$, the Laplace transform of the two-point function in the direction $s$ is infinite for $β=β_{\text{sat}}(s)$ (where $β_{\text{sat}}(s)$ is a the biggest value such that the inverse correlation length $ν_β(s)$ associated to the truncated two-point function is equal to $ρ(s)$ on $[0,β_{\text{sat}}(s)))$. Moreover, we prove that the two-point function satisfies Ornstein-Zernike asymptotics for $β=β_{\text{sat}}(s)$ on $\mathbb{Z}$. As far as we know, this constitutes the first result on the behaviour of the two-point function at $β_{\text{sat}}(s)$. Finally, we show that there exists $β_{0}$ such that for every $β>β_{0}$, $ν_β(s)=ρ(s)$. All the results are new.

math.PR↗