WIAS Preprint No. 1532, (2010)

Self-intersection local times of random walks: Exponential moments in subcritical dimensions


  • Becker, Mathias
  • König, Wolfgang
    ORCID: 0000-0002-7673-4364

2010 Mathematics Subject Classification

  • 60K37 60F10 60J55


  • Self-intersection local time, upper tail, Donsker-Varadhan large deviations, variational formula, Gagliardo-Nirenberg inequality




Fix $p>1$, not necessarily integer, with $p(d-2)0$ that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of $t$ and $theta_t$, and the precise rate is characterized in terms of a variational formula, which is in close connection to the it Gagliardo-Nirenberg inequality. As a corollary, we obtain a large-deviation principle for $ ell_t _p/(t r_t)$ for deviation functions $r_t$ satisfying $t r_tggE[ ell_t _p]$. Informally, it turns out that the random walk homogeneously squeezes in a $t$-dependent box with diameter of order $ll t^1/d$ to produce the required amount of self-intersections. Our main tool is an upper bound for the joint density of the local times of the walk.

Appeared in

  • Probab. Theory Related Fields, 154 (2012) pp. 585--605.

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