Crossover from subcritical to critical decay: random walk, self-avoiding walk, percolation
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910223448408064 |
|---|---|
| author | Liu, Yucheng Slade, Gordon |
| author_facet | Liu, Yucheng Slade, Gordon |
| contents | The study of the Ornstein--Zernike decay of subcritical two-point functions in equilibrium statistical mechanics has a history going back over a century. Despite this, the crossover from Ornstein--Zernike decay to critical power-law decay has received scant attention in the literature. We prove a general theorem which, under appropriate hypotheses, identifies the asymptotic behaviour of the solution to an Ornstein--Zernike equation on $\mathbb{Z}^d$ as that of the Green function for Brownian motion with drift, multiplied by an anisotropic exponentially decaying factor. The theorem applies to a wide class of random walks, to nearest-neighbour self-avoiding walk in dimensions $d \ge 5$, and to nearest-neighbour percolation in dimensions $d \ge 15$. Wide-ranging consequences follow, including details of the crossover from Ornstein--Zernike to critical decay on the scale of the correlation length, and the fact that all finite-order correlation lengths are equivalent up to universal constants. The proof is based on a variational characterisation of the direction-dependent rate of exponential decay and a major extension of Hara's 2008 Gaussian Lemma to noncentred kernels. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15545 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Crossover from subcritical to critical decay: random walk, self-avoiding walk, percolation Liu, Yucheng Slade, Gordon Probability Mathematical Physics 60K35, 82B27, 82B41, 82B43 The study of the Ornstein--Zernike decay of subcritical two-point functions in equilibrium statistical mechanics has a history going back over a century. Despite this, the crossover from Ornstein--Zernike decay to critical power-law decay has received scant attention in the literature. We prove a general theorem which, under appropriate hypotheses, identifies the asymptotic behaviour of the solution to an Ornstein--Zernike equation on $\mathbb{Z}^d$ as that of the Green function for Brownian motion with drift, multiplied by an anisotropic exponentially decaying factor. The theorem applies to a wide class of random walks, to nearest-neighbour self-avoiding walk in dimensions $d \ge 5$, and to nearest-neighbour percolation in dimensions $d \ge 15$. Wide-ranging consequences follow, including details of the crossover from Ornstein--Zernike to critical decay on the scale of the correlation length, and the fact that all finite-order correlation lengths are equivalent up to universal constants. The proof is based on a variational characterisation of the direction-dependent rate of exponential decay and a major extension of Hara's 2008 Gaussian Lemma to noncentred kernels. |
| title | Crossover from subcritical to critical decay: random walk, self-avoiding walk, percolation |
| topic | Probability Mathematical Physics 60K35, 82B27, 82B41, 82B43 |
| url | https://arxiv.org/abs/2605.15545 |