BALLISTICITY CONDITIONS FOR RANDOM WALK IN
RANDOM ENVIRONMENT
ALEJANDRO F. RAMIREZ
resumen. Consider a Random Walk in a Random Environment (RWRE)
{Xn :
n
≥ 0} on a uniformly elliptic i.i.d. environment in dimensions d ≥ 2. Some
fundamental questions about this model, related to the concept of ballisticity
and which remain unsolved, will be discussed in this talk. The walk is said to
be transient in a direction l
∈ S
d
, if limn
→∞ Xn l = ∞, and ballistic in the
direction l if lim inf n
→∞ Xn l/n > 0. It is conjectured that transience in a
given direction implies ballisticity in the same direction. To tackle this question,
in 2002, Sznitman introduced for each γ
∈ (0, 1) and direction l the ballisticity
condition (Tγ )
|l, and condition (T ′ )|l defined as the fulfillment of (Tγ )|l for each
γ
∈ (0, 1). He proved that (T ′ ) implies ballisticity in the corresponding direction,
and showed that for each γ
∈ (0, 5, 1), (Tγ ) implies (T ′ ). It is believed that for
each γ
∈ (0, 1), (Tγ ) implies (T ′ ). We prove that for γ ∈ (γd , 1), (T )γ is equivalent
to (T ′ ), where for d
≥ 4, γd = 0 while for d = 2, 3 we have γd ∈ (0.366, 0.388).
The case d
≥ 4 uses heavily a recent multiscale renormalization method developed
by Noam Berger. This talk is based on joint works with Alexander Drewitz from
ETH Z
urich.
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