On the speed of once-reinforced biased random walk on trees

Abstract

We study the asymptotic behaviour of once-reinforced biased random walk (ORbRW) on Galton-Watson trees. Here the underlying (unreinforced) random walk has a bias towards or away from the root. We prove that in the setting of multiplicative once-reinforcement the ORbRW can be recurrent even when the underlying biased random walk is ballistic. We also prove that, on Galton-Watson trees without leaves, the speed is positive in the transient regime. Finally, we prove that, on regular trees, the speed of the ORbRW is monotone decreasing in the reinforcement parameter when the underlying random walk has high speed, and the reinforcement parameter is small.The proof of Theorem 1.12 is inspired by work of Ben Arous, Fribergh and Sidoravicius [4], who proved a partial monotonicity result for biased random walks on Galton-Watson trees via a nice and natural coupling. Here we needed to adapt this idea to find a coupling which works for certain self-interacting walks. We have taken some care to extract the general features of the argument (which are relatively simple) and the details of the coupling for our particular setting (which are highly non-trivial). This general method provides a coupling alternative to expansion techniques (e.g. [10]) when the self-interacting random walk has a large bias independent of the history. The modelConsider a rooted tree G = (V, E) (where V is the set of vertices and E is the set of edges), augmented by adding a parent −1 to the root , the two being connected by an edge. If two vertices ν and µ are the endpoints of the same edge, they are said to be neighbours, and this property is denoted by ν ∼ µ. The distance d(ν, µ) between any pair of vertices ν, µ, not necessarily adjacent, is the number of edges in the unique self-avoiding path connecting ν to µ. We set | −1 | = −1. For any other vertex ν, we let |ν| be the distance from ν to the root . We use ν −1 to denote the parent of ν, and (ν i ) i∈[∂(ν)] , its children, where ∂(ν) is the number of offspring of ν and [n] := {1, 2, . . . , n}. Write C ν = {ν 1 , . . . , ν ∂(ν) }. For µ ∈ V \ { −1 } we write ν < µ, if ν is an ancestor of µ, i.e. ν lies on the self-avoiding path connecting µ to −1 , and we write ν ≤ µ if ν < µ or ν = µ. If ν < µ then µ is said to be a descendant of ν.We are now about to define MAD walks, standing for maximum acts differently, in reference to their behavior on Z. Fix G = (V, E) and parameters u 1 , u 0 > 0 and define the law P G of a MAD walk X = (X n ) n∈Z + , taking values on the vertices of G (and taking nearest neighbour steps) as follows. We set X 0 = −1 . Given F n = σ(X k : k ≤ n), we let E ∅ (n) = {[X k−1 , X k ] : k ≤ n} ⊂ E denote the set of (undirected) edges crossed by time n.For ν = −1 define W n (ν, ν −1 ) = 1, (1.1)

The Branching‐Ruin Number and the Critical Parameter of Once‐Reinforced Random Walk on Trees

“…A process X is a biased ORRW with parameter δ ∈ (0, ∞) if it is a GORW with initial weights w e = β −|e| and reinforced weights δ e = δ × β −|e| for every edge e ∈ E. The case β > 1 corresponds to a bias towards the root and the case β ∈ (0, 1) corresponds to an outward bias. The next result generalizes Corollary 1.5 in [6]. Note that the case δ = 1 corresponds to a usual biased random walk, and the case β = 1 corresponds to ORRW.…”

“…Here, we define the same construction as in [6] which is a particular case of Rubin's construction. This will allow us to emphasize useful independence properties of the walk on disjoint subsets of the tree.…”

“…As it was done in [6], we are now going to define a family of coupled processes on the subtrees of G. For any rooted subtree G of G, Let us define the extension X (G ) = (V , E ) on G as follows. Let the root of G be defined as the vertex of V with smallest distance to .…”

“…Durrett, Kesten and Limic [10] proved that this conjecture does not hold on the binary tree and that ORRW is transient for any choice of parameter. This was extended to supercritical Galton-Watson trees in [4] (see also [6] where the positivity and the monotonicity of the speed on Galton-Watson trees is studied). Some partial results on ladders [20,22] are also available.…”

“…The case when w e = 1 and δ e = δ for any e ∈ E and for some δ ∈ (0, ∞) corresponds to the Once edge-reinforced random walk (ORRW) with parameter δ. Note that the model we defined includes usual reversible Markov chains on trees, as well as various generalized versions of the ORRW (see [6] for instance).…”