Abstract
Fix p > 1, not necessarily integer, with p(d − 2) < d. We study the p-fold self-intersection local time of a simple random walk on the lattice \({\mathbb Z^d}\) up to time t. This is the p-norm of the vector of the walker’s local times, ℓ t . We derive precise logarithmic asymptotics of the expectation of exp{θ t ||ℓ t || p } for scales θ t > 0 that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of t and θ t , and the precise rate is characterized in terms of a variational formula, which is in close connection to the Gagliardo–Nirenberg inequality. As a corollary, we obtain a large-deviation principle for ||ℓ t || p /(tr t ) for deviation functions r t satisfying \({t r_t\gg \mathbb E[||\ell_t||_p]}\) . Informally, it turns out that the random walk homogeneously squeezes in a t-dependent box with diameter of order ≪ 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.
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Becker, M., König, W. Self-intersection local times of random walks: exponential moments in subcritical dimensions. Probab. Theory Relat. Fields 154, 585–605 (2012). https://doi.org/10.1007/s00440-011-0377-0
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DOI: https://doi.org/10.1007/s00440-011-0377-0
Keywords
- Self-intersection local time
- Upper tail
- Donsker–Varadhan large deviations
- Variational formula
- Gagliardo–Nirenberg inequality
Mathematics Subject Classification (2000)
- 60K37
- 60F10
- 60J55