We show that the contact process on a random d-regular graph initiated by a single infected vertex obeys the “cutoff phenomenon” in its supercritical phase. In particular, we prove that, when the infection rate is larger than the lower critical value of the contact process on the infinite d-regular tree, there are positive constants C, p depending on the infection rate such that for any ε > 0, ...

General Relativity is the study of Lorentz 4-manifolds (M, g) whose metric arises from a solution to the Einstein equations. We will take the signature of the metric to be (− + ++), where the minus sign indicates the time component. Then the tangent space at any point may be divided into three different types of vectors, namely timelike, spacelike, and null vectors whose lengths squared respect...

Let T be a d-regular tree (d ≥ 3) and A = Aut(T ) its automorphism group. Let Γ be the group generated by n independent Haar-random elements of A. We show that almost surely, every nontrivial element of Γ has finitely many fixed points on T .

Given a bipartite connected finite graph G=(V, E) and a vertex v0∈V, we consider a uniform probability measure on the set of graph homomorphisms f: V→Z satisfying f(v0)=0. This measure can be viewed as a Gindexed random walk on Z, generalizing both the usual time-indexed random walk and tree-indexed random walk. Several general inequalities for the G-indexed random walk are derived, including a...

The Stone tautologies are known to have polynomial size resolution refutations and require exponential size regular refutations. We prove that the Stone tautologies also have polynomial size proofs in both pool resolution and the proof system of regular tree-like resolution with input lemmas (regRTI). Therefore, the Stone tautologies do not separate resolution from DPLL with clause learning.

We establish that the phase transition for infinite cycles in the random stirring model on an infinite regular tree of high degree is sharp. That is, we prove that there exists d0 such that, for any d ≥ d0, the set of parameter values at which the random stirring model on the rooted regular tree with offspring degree d almost surely contains an infinite cycle consists of a semi-infinite interva...

We consider verification of programs manipulating dynamic linked data structures such as various forms of singly and doubly-linked lists or trees. We consider important properties for this kind of systems like no null-pointer dereferences, absence of garbage, shape properties, etc. We develop a verification method based on a novel use of tree automata to represent heap configurations. A heap is...

Recently, the question has been raised whether the derivation tree languages of Minimalist grammars (MGs; [14, 16]) are closed under intersection with regular tree languages [4, 5]. Using a variation of a proof technique devised by Thatcher [17], I show that even though closure under intersection does not obtain, it holds for every MG and regular tree language that their intersection is identic...

When dealing with infinite-state systems, Regular Tree Model Checking approaches may have some difficulties to represent infinite sets of data. We propose Lattice Tree Automata, an extended version of tree automata to represent complex data domains and their related operations in an efficient manner. Moreover, we introduce a new completionbased algorithm for computing the possibly infinite set ...

In this paper, we extend the notion of tree language quotients to bottom-up quotients. Instead of computing the residual of a tree language from top to bottom and producing a list of tree languages, we show how to compute a set of k-ary trees, where k is an arbitrary integer. We define the quotient formula for different combinations of tree languages: union, symbol products, compositions, itera...