Localisation and delocalisation of the integer-valued membrane model
Abstract
We study the integer-valued membrane model on boxes [−L,L]d∩ℤd with zero boundary conditions. We show that in dimension two the model is delocalised at sufficiently high temperature. More precisely, the variance at the origin is of order L2/β, where β is the inverse temperature. More generally, for every ε>0 and all sufficiently small β>0, the logarithm of each Laplace transform lies between (1+ε)−1 and $1$ times its Gaussian counterpart. The temperature threshold is independent of L and the test function. In dimensions three and four, we prove that the variance at the origin is bounded uniformly in the box size at every positive inverse temperature. Thus the integer constraint changes the order of fluctuations in these dimensions. The proofs use a modification of the Fröhlich--Spencer argument in dimension two and a comparison with integer-valued Gaussian free fields in higher dimensions.
Related
- On the Krull dimensions of rings of integer-valued polynomials and polynomial rings
- Stacked-bases Properties, maximal ideals and polynomial closures of Rings of Integer-Valued Polynomials
- Integer-valued polynomials over structural matrix rings
- Cyclotomic Newton Expansions and a Rank-Uniform Integer-Valued Newton Completion
- Precise Delocalisation and Gumbel Laws for Eigenvectors of Wigner Matrices
BibTeX
@misc{cipriani2026localisation,
title = {{Localisation and delocalisation of the integer-valued membrane model}},
author = {Alessandra Cipriani and Biltu Dan and Rajat Subhra Hazra},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41925},
eprint = {2610.41925},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41925},
}
RIS
TY - UNPB
TI - Localisation and delocalisation of the integer-valued membrane model
AU - Alessandra Cipriani
AU - Biltu Dan
AU - Rajat Subhra Hazra
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41925
UR - https://arxiv.org/abs/2610.41925
AB - We study the integer-valued membrane model on boxes [−L,L]^d∩ℤ^d with zero boundary conditions. We show that in dimension two the model is delocalised at sufficiently high temperature. More precisely, the variance at the origin is of order L^2/β, where β is the inverse temperature. More generally, for every ε>0 and all sufficiently small β>0, the logarithm of each Laplace transform lies between (1+ε)^−1 and $1$ times its Gaussian counterpart. The temperature threshold is independent of L and the test function. In dimensions three and four, we prove that the variance at the origin is bounded uniformly in the box size at every positive inverse temperature. Thus the integer constraint changes the order of fluctuations in these dimensions. The proofs use a modification of the Fröhlich--Spencer argument in dimension two and a comparison with integer-valued Gaussian free fields in higher dimensions.
ER -