An isomorphism between the p-adic integers and a ring associated with a tiling of N-space by permutohedra

AbstractThe classical lattice A∗n, whose Voronoi cells tile Euclidean n-space by permutohedra, can be given the generalized balance ternary ring structure GBTn in a natural way as a quotient ring of Z[x]. The ring GBTn can also be considered as the set of all finite sequences s0 s1…sk, with si ∈ GBTn⧸αGBTn for all i, where α is an appropriately chosen element in GBTn. The extended generalized balance ternary (EGBTn) ring consists of all such infinite sequences. A primary goal of this paper is to prove that if 2n+1−1 and n+1 are relatively prime, then EGBTn is isomorphic as a ring to the (2n+1−1)-adic integers

Tricolor percolation and random paths in 3D

We study "tricolor percolation" on the regular tessellation of R[superscript 3] by truncated octahedra, which is the three-dimensional analog of the hexagonal tiling of the plane. We independently assign one of three colors to each cell according to a probability vector p = (p1,p2,p3) and define a "tricolor edge" to be an edge incident to one cell of each color. The tricolor edges form disjoint loops and/or infinite paths. These loops and paths have been studied in the physics literature, but little has been proved mathematically. We show that each p belongs to either the compact phase (in which the length of the tricolor loop passing through a fixed edge is a.s. finite, with exponentially decaying law) or the extended phase (in which the probability that an (n × n × n) box intersects a tricolor path of diameter at least n exceeds a positive constant, independent of n). We show that both phases are non-empty and the extended phase is a closed subset of the probability simplex. We also survey the physics literature and discuss open questions, including the following: Does p = (1/3,1/3,1/3) belong to the extended phase? Is there a.s. an infinite tricolor path for this p? Are there infinitely many? Do they scale to Brownian motion? If p lies on the boundary of the extended phase, do the long paths have a scaling limit analogous to SLE6 in two dimensions? What can be shown for the higher dimensional analogs of this problem?United States-Israel Binational Science Foundation (Grant 2010357)National Science Foundation (U.S.) (Grant DMS 064558)National Science Foundation (U.S.) (Grant 1209044