Search ResultsShowing 1-6 of 6
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arXiv:2309.04454 (Published 2023-09-08)
An upper bound on geodesic length in 2D critical first-passage percolation
Comments: 62 pages, 14 figuresWe consider i.i.d. first-passage percolation (FPP) on the two-dimensional square lattice, in the critical case where edge-weights take the value zero with probability $1/2$. Critical FPP is unique in that the Euclidean lengths of geodesics are superlinear, rather than linear, in the distance between their endpoints. This fact was speculated by Kesten in 1986 but not confirmed until 2019 by Damron and Tang, who showed a lower bound on geodesic length that is polynomial with degree strictly greater than $1$. In this paper we establish the first non-trivial upper bound. Namely, we prove that for a large class of critical edge-weight distributions, the shortest geodesic from the origin to a box of radius $R$ uses at most $R^{2+\epsilon}\pi_3(R)$ edges with high probability, for any $\epsilon > 0$. Here $\pi_3(R)$ is the polychromatic 3-arm probability from classical Bernoulli percolation; upon inserting its conjectural asymptotic, our bound converts to $R^{4/3 + \epsilon}$. In any case, it is known that $\pi_3(R) \lesssim R^{-\delta}$ for some $\delta > 0$, and so our bound gives an exponent strictly less than $2$. In the special case of Bernoulli($1/2$) edge-weights, we replace the additional factor of $R^\epsilon$ with a constant and give an expectation bound.
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arXiv:2211.14365 (Published 2022-11-25)
A dichotomy theory for height functions
Comments: 65 pages, 17 figuresHeight functions are random functions on a given graph, in our case integer-valued functions on the two-dimensional square lattice. We consider gradient potentials which (informally) lie between the discrete Gaussian and solid-on-solid model (inclusive). The phase transition in this model, known as the roughening transition, Berezinskii-Kosterlitz-Thouless transition, or localisation-delocalisation transition, was established rigorously in the 1981 breakthrough work of Fr\"ohlich and Spencer. It was not until 2005 that Sheffield derived continuity of the phase transition. First, we establish sharpness, in the sense that covariances decay exponentially in the localised phase. Second, we show that the model is delocalised at criticality, in the sense that the set of potentials inducing localisation is open in a natural topology. Third, we prove that the pointwise variance of the height function is at least $c\log n$ in the delocalised regime, where $n$ is the distance to the boundary, and where $c>0$ denotes a universal constant. This implies that the effective temperature of any potential cannot lie in the interval $(0,c)$ (whenever it is well-defined), and jumps from $0$ to at least $c$ at the critical point. We call this range of forbidden values the effective temperature gap.
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arXiv:1701.04482 (Published 2017-01-16)
Asymptotic behaviour of ground states for mixtures of ferromagnetic and antiferromagnetic interactions in a dilute regime
We consider randomly distributed mixtures of bonds of ferromagnetic and antiferromagnetic type in a two-dimensional square lattice with probability $1-p$ and $p$, respectively, according to an i.i.d. random variable. We study minimizers of the corresponding nearest-neighbour spin energy on large domains in ${\mathbb Z}^2$. We prove that there exists $p_0$ such that for $p\le p_0$ such minimizers are characterized by a majority phase; i.e., they take identically the value $1$ or $-1$ except for small disconnected sets. A deterministic analogue is also proved.
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arXiv:1501.02966 (Published 2015-01-13)
Some results and problems for anisotropic random walks on the plane
Comments: 20 pagesCategories: math.PRThis is an expository paper on the asymptotic results concerning path behaviour of the anisotropic random walk on the two-dimensional square lattice Z^2. In recent years Mikl\'os and the authors of the present paper investigated the properties of this random walk concerning strong approximations, local times and range. We give a survey of these results together with some further problems.
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Phase transitions in layered systems
Luiz Renato Fontes, Domingos H. U. Marchetti, Immacolata Merola, Errico Presutti, Maria Eulalia VaresComments: 17 pages. Final version. Published in Journal of Statistical Physics (2014), volume 157, 407-421Journal: J Stat Phys 157, 2014, 407-421Categories: math.PRKeywords: phase transition, layered systems, nearest neighbor ferromagnetic vertical interaction, mean field critical value, two-dimensional square latticeTags: journal articleWe consider the Ising model on the two-dimensional square lattice where on each horizontal line, called "layer", the interaction is given by a ferromagnetic Kac potential with coupling strength $J_\gamma(x,y)=\gamma J(\gamma(x-y))$, where $J(\cdot)$ is smooth and has compact support; we then add a nearest neighbor ferromagnetic vertical interaction of strength $\gamma^{A}$ (where $A\ge 2$ is fixed) and prove that for any $\beta$ (inverse temperature) larger than the mean field critical value there is a phase transition for all $\gamma$ small enough.
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arXiv:1307.7222 (Published 2013-07-27)
Incipient infinite cluster in 2D Ising percolation
Comments: 8 pagesCategories: math.PRWe consider the percolation problem in the high-temperature Ising model on the two-dimensional square lattice at or near critical external fields. The incipient infinite cluster (IIC) measure in the sense of Kesten is constructed. As a consequence, we can obtain some geometric properties of IIC. The result holds also for the triangular lattice.