Capacity of the range of branching random walks in low dimensions

Abstract: Consider a branching random walk

(V_{u})_{u i n m a t h c a l T^{I G W}}

in

m a t h b b Z^{d}

with the genealogy tree

m a t h c a l T^{I G W}

formed by a sequence of i.i.d. critical Galton-Watson trees. Let

R_{n}

be the set of points in

m a t h b b Z^{d}

visited by

(V_{u})

when the index

u

explores the first

n

subtrees in

m a t h c a l T^{I G W}

. Our main result states that for

d i n 3, 4, 5

, the capacity of

R_{n}

is almost surely equal to

n^{f r a c d - 2 2 + o (1)}

as

n o i n f t y

.

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