Figure 3 - uploaded by Dante Mata
Content may be subject to copyright.
Illustration of the decomposition into edge-disjoint trees t A , t B , t C , t D appearing in Lemma 3.2 (2δn and δn are not to scale). [Figure reused by permission of Springer, from Commun. Math. Phys. 331 (2014), 67-109 (Electrical resistance of the low-dimensional critical branching random walk. A.A. Járai and A. Nachmias) c (2014).]
Source publication
We study the trace of the incipient infinite oriented branching random walk in $\mathbb{Z}^d \times \mathbb{Z}_+$ when the dimension is $d = 6$. Under suitable moment assumptions, we show that the electrical resistance between the root and level $n$ is $O(n \log^{-\xi}n )$ for a $\xi > 0$ that does not depend on details of the model.
Contexts in source publication
Context 1
... assume that Z is at height k 1 . Given a tree t ⊂ T M such that V, U ∈ t and V does not have any children in t, we have a unique decomposition of t into edge disjoint trees (t A , ρ, Z), (t B , Z + , U ), t C and t D , see Figure 3. The doubly rooted tree (t A , ρ, Z) contains all the descendants of ρ that are not descendants of Z. ...
Context 2
... assume that Z is at height k 1 . Given a tree t ⊂ T M such that V, U ∈ t and V does not have any children in t, we have a unique decomposition of t into edge disjoint trees (t A , ρ, Z), (t B , Z + , U ), t C and t D , see Figure 3. The doubly rooted tree (t A , ρ, Z) contains all the descendants of ρ that are not descendants of Z. ...