A sufficient condition for a graph to be weakly k-linked

AbstractFor a pair (s, t) of vertices of a graph G, let λG(s, t) denote the maximal number of edge-disjoint paths between s and t. Let (s1, t1), (s2, t2), (s3, t3) be pairs of vertices of G and k > 2. It is shown that if λG(si, ti) ≥ 2k + 1 for each i = 1, 2, 3, then there exist 2k + 1 edge-disjoint paths such that one joins s1 and t1, another joins s2 and t2 and the others join s3 and t3. As a corollary, every (2k + 1)-edge-connected graph is weakly (k + 2)-linked for k ≥ 2, where a graph is weakly k-linked if for any k vertex pairs (si, ti), 1 ≤ i ≤ k, there exist k edge-disjoint paths P1, P2,…, Pk such that Pi joins si and ti for i = 1, 2,…, k

Weak Decoherence and Quantum Trajectory Graphs

Griffiths' ``quantum trajectories'' formalism is extended to describe weak decoherence. The decoherence conditions are shown to severely limit the complexity of histories composed of fine-grained events.Comment: 12 pages, LaTeX, 3 figures (uses psfig), all in a uuencoded compressed tar fil

Coning-off CAT(0) cube complexes

In this paper, we study the geometry of cone-offs of CAT(0) cube complexes over a family of combinatorially convex subcomplexes, with an emphasis on their Gromov-hyperbolicity. A first application gives a direct cubical proof of the characterization of the (strong) relative hyperbolicity of right-angled Coxeter groups, which is a particular case of a result due to Behrstock, Caprace and Hagen. A second application gives the acylindrical hyperbolicity of $C'(1/4)-T(4)$ small cancellation quotients of free products.Comment: 45 pages, 13 figures. Comments are welcom

Percolation on an infinitely generated group

We give an example of a long range Bernoulli percolation process on a group non-quasi-isometric with $\mathbb{Z}$, in which clusters are almost surely finite for all values of the parameter. This random graph admits diverse equivalent definitions, and we study their ramifications. We also study its expected size and point out certain phase transitions.Comment: 23 page

Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits

We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of for00504925mulas and weakly skew circuits. Our representations produce matrices of much smaller dimensions than those given in the convex geometry literature when applied to polynomials having a concise representation (as a sum of monomials, or more generally as an arithmetic formula or a weakly skew circuit). These representations are valid in any field of characteristic different from 2. In characteristic 2 we are led to an almost complete solution to a question of B\"urgisser on the VNP-completeness of the partial permanent. In particular, we show that the partial permanent cannot be VNP-complete in a finite field of characteristic 2 unless the polynomial hierarchy collapses.Comment: To appear in the AMS Contemporary Mathematics volume on Randomization, Relaxation, and Complexity in Polynomial Equation Solving, edited by Gurvits, Pebay, Rojas and Thompso