On large‐girth regular graphs and random processes on trees

Abstract

We study various classes of random processes defined on the regular tree T d that are invariant under the automorphism group of T d . The most important ones are factor of i.i.d. processes (randomized local algorithms), branching Markov chains and a new class that we call typical processes. Using Glauber dynamics on processes we give a sufficient condition for a branching Markov chain to be factor of i.i.d. K E Y W O R D Sfactor of i.i.d.

The distribution of sandpile groups of random regular graphs

“…This means that if we choose H n independently, then with probability 1 the sequence converges. Following the lines of Theorem 7 one can prove that lim n→∞ max p is a prime co-rank p (H n ) n = 0 with probability 1, which settles this special case of the conjecture, and we even get a uniform convergence in p. Note that this has been proved by Backhausz and Szegedy [2] using a different method.…”

“…Observe that M(q, r) consists of the non-negative integral points of a certain affine subspace A(q, r) of R V h . This affine subspace A(q, r) is determined by linear equations expressing that whenever Σ(Q) = r for a (q, h)-tuple Q = (q (1) , q (2) , . .…”

“…Let H 0 = {j | r q (j) ∈ dW }. For i = 1, 2, ..., d − 1 we define the random subset H i of {1, 2, ..., n} using the random (q, d − 1)-tupleQ = (q (1) ,q (2) , . .…”

“…, q n is denoted by MinC q , the sum of the components of q is denoted by s(q) = n i=1 q i , and we define R(q, d) = {r ∈ (d · MinC q ) n | s(r) = ds(q)}. 2 It is straightforward to check that A n q ∈ R(q, d) with probability 1. Let U q,d be a uniform random element of R(q, d).…”

Total Domination in Regular Graphs

“…For d ≥ 3, a graph G ∈ G n,d asymptotically almost surely satisfies γ t (G)/n ≤ Γ g d for the values of Γ g d given in Table 1. In fact, the connection between the behavior of graph parameters for graphs with large girth and for random regular graphs given in (5) in the context of total domination actually holds for many different parameters, and it is a significant open question whether (5) holds with equality (see Backhausz and Szegedy [3] for a detailed description of problems in this line of research). Recently, Wormald and the first author [17] proved that an upper bound on γ g t (d, ∞) implies an upper bound on γ R t (d) as long as it is obtained through the analysis of a local algorithm, as described in their paper.…”