Is critical 2D percolation universal?

Vincent Beffara

Home/Archive/arXiv

arXiv · 0708.3908

Is critical 2D percolation universal?

Vincent Beffara

View original source ↗Read source document ↗Export metadata ↓

Abstract

The aim of this paper is to explore possible ways of extending Smirnov's proof of Cardy's formula for critical site-percolation on the triangular lattice to other cases (such as bond-percolation on the square lattice); the main question we address is that of the choice of the lattice embedding into the plane which gives rise to conformal invariance in the scaling limit. Even though we were not able to produce a complete proof, we believe that the ideas presented here go in the right direction.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Vincent Beffara. 2007-08-29. Is critical 2D percolation universal?. https://arxiv.org/abs/0708.3908

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the First Hitting Time Problems for Diffusion Processes: Local Time-Space Approach

Using the local time-space calculus of Peskir (2005) and the method developed in Mijatovic (2010), we derive a new integral representation for the backward probability associated with the first-passage time (FPT) of a diffusion process through a time-dependent barrier. We present a complete three-step numerical algorithm: first, the problem is reduced to a Volterra-type integral equation; second, its kernel is approximated by a Markov chain; finally, the resulting equation is solved using a quadrature rule. The method is implemented for several representative examples, and its convergence properties are established. An extension to double barrier problems is carried out

math.PR↗

arXiv2026

Markovian Random Words, a Gaussian Random Matrix Model and Beyond

After centering and scaling, the length of the longest weakly increasing subsequence of a finite-state Markov random word converges to a Brownian last-passage functional with dependent components. We investigate when this functional, and more generally the limiting RSK Young-diagram shape, admits a Gaussian random-matrix representation. We introduce Hermitian Gaussian matrices with correlated diagonal entries and show that isotropy of the Brownian covariance on the trace-zero hyperplane yields such a representation. Within this natural class of Hermitian models with independent Gaussian off-diagonal entries, we prove a rigidity theorem showing that isotropy, together with equality of the off-diagonal variances, characterizes the conditional largest-eigenvalue representation. Explicit three-letter non-cyclic Markov chains demonstrate that this representation is not universal: the limiting Brownian functional may fail to have the law of the largest eigenvalue of any centered jointly Gaussian $3\times 3$ Hermitian matrix. Finally, we extend the random-matrix identification to chains with non-uniform stationary distributions and to countable state spaces.

math.PR↗

arXiv2026

The Order Mass Identity of Rooted Forests, with a Direct Proof of the Extended Aldous--Broder Theorem

Random spanning forests provide probability models for partitions and Monte Carlo methods on graphs. In these applications, the distribution produced by a sampler determines which forest functionals can be estimated. The Aldous--Broder algorithm constructs a spanning tree from the first-entrance edges of a covering random walk. Its distribution is known, but a direct calculation requires summing over all orders in which a fixed tree can be discovered. We evaluate this sum through an identity for rooted forests and real matrices with zero row sums. Under a nonsingularity condition on the principal submatrices encountered during growth, the sum of products of harmonic extension entries equals a principal cofactor of a matrix on the roots. With one root, the sum is one. The identity gives a direct proof of the extended Aldous--Broder theorem for every finite irreducible Markov chain started at a fixed state, without assuming reversibility or bidirectional transition support. It also gives the conditional distribution of the discovery order and the law of the forest obtained by stopping after all nonroot vertices have been visited. A completion weight describes the difference between this stopped-walk law and the weight-proportional forest law, including the support restriction needed for importance correction. Numerical examples illustrate the sampling implications. The identity, the Aldous--Broder theorem, and the stopped-forest law are formalized in Lean4.

math.PR↗