Related papers: Nested Closed Paths in Two-Dimensional Percolation

A useful result about leftmost and rightmost paths in two dimensional bond percolation is proved. This result was introduced without proof in \cite{G} in the context of the contact process in continuous time. As discussed here, it also…

Using a recently developed method to simulate percolation on large clusters of distributed machines [N. R. Moloney and G. Pruessner, Phys. Rev. E 67, 037701 (2003)], we have numerically calculated crossing, spanning and wrapping…

Let $\{X(v), v \in \Bbb Z^d \times \Bbb Z_+\}$ be an i.i.d. family of random variables such that $P\{X(v)= e^b\}=1-P\{X(v)= 1\} = p$ for some $b>0$. We consider paths $\pi \subset \Bbb Z^d \times \Bbb Z_+$ starting at the origin and with…

We prove a Russo-Seymour-Welsch percolation theorem for nodal domains and nodal lines associated to a natural infinite dimensional space of real analytic functions on the real plane. More precisely, let $U$ be a smooth connected bounded…

This is the first of two articles devoted to the analysis of the path description of the states in su(2)_k WZW models, a representation well suited for constructive derivations of the fermionic characters. In this first article, the cases…

Using conformal field theory, we derive several new crossing formulas at the two-dimensional percolation point. High-precision simulation confirms these results. Integrating them gives a unified derivation of Cardy's formula for the…

The state space of our model is the Euclidean space in dimension d = 2. Simultaneously, from all points of a homogeneous Poisson point process, we let grow independent and identically distributed random continuum paths. Each path stops…

We study the number $N\_n$ of open paths of length $n$ in supercritical oriented percolation on $\Zd \times \N$, with $d \ge 1$. We prove that on the percolation event $\{\inf N\_n\textgreater{}0\}$, $N\_n^{1/n}$ almost surely converges to…

The path-following scheme in [Loisel and Maxwell, SIAM J. Matrix Anal. Appl., 39-4 (2018), pp. 1726-1749] is adapted to efficiently calculate the dispersion relation curve for linear surface waves on an arbitrary vertical shear current.…

We conjecture a combinatorial formula for the monomial expansion of the image of any Schur function under the Bergeron-Garsia nabla operator. The formula involves nested labeled Dyck paths weighted by area and a suitable "diagonal…

The k-neighbor graph is a directed percolation model on the hypercubic lattice Z d in which each vertex independently picks exactly k of its 2d nearest neighbors at random, and we open directed edges towards those. We prove that the…

Aim of this paper is to prove the second order differentiation formula along geodesics in compact $RCD^*(K,N)$ spaces with $N<\infty$. This formula is new even in the context of Alexandrov spaces. We establish this result by showing that…

The vacuum polarization of a quark, when considered in terms of the external momentum q^2, is a function of the Stieltjes type. Consequently, the mathematical theory of Pade Approximants assures that the full function, at any finite value…

We construct exact eigenvectors and eigenvalues for $U_q(\mathfrak{sp}_{2n})$- and $U_q(\mathfrak{so}_{2n})$-symmetric closed spin chains by means of a nested algebraic Bethe ansatz method. We use a fusion procedure to construct…

Combining probabilistic and analytic tools from potential theory, we investigate Dirichlet problems associated with the Dunkl Laplacian $\Delta_k$. We establish, under some conditions on the open set $D\subset\R^d$, the existence of a…

We present an exact, closed-form expression for the Newtonian potential of the characteristic function associated with two overlapping discs in the plane. This setting naturally arises when discretising nonlocal interaction terms present in…

We present a fine-grained approach to identify clusters and perform percolation analysis in a 2D lattice system. In our approach, we develop an algorithm based on the linked-list data structure whereby the members of a cluster are nodes of…

We study the behavior of the optimal path between two sites separated by a distance $r$ on a $d$-dimensional lattice of linear size $L$ with weight assigned to each site. We focus on the strong disorder limit, i.e., when the weight of a…

The Neumann-Poincar\'e (NP) operator naturally appears in the context of metamaterials as it may be used to represent the solutions of elliptic transmission problems via potentiel theory. In particular, its spectral properties are closely…

Factorization method is developed for a family of discretely spiked harmonic oscillators. Two sets of intertwining and ladder operators are presented to algebraically generate eigenstates with energies isomorphic to those of the ordinary…