Boundary of the Range I: Typical Behavior

Amine Asselah (1) , Bruno Schapira (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 I2M - Institut de Mathématiques de Marseille

We study the boundary of the range for the simple random walk on Z d in the transient regime d ≥ 3. We show that sizes of the range and its boundary differ mainly by a martingale. As a consequence, we obtain a bound on the variance of order n log n in dimension three. We also establish a central limit theorem in dimension four and larger.

Mots clés: boundary range; random walk; central limit theorem

Domaines: Probabilités [math.PR]

Amine Asselah, Bruno Schapira. Boundary of the Range I: Typical Behavior. 2015. ⟨hal-01171542v2⟩