Abstract
Random Walk conditioned to stay above a non-flat floor: curvature effects
S. Ott and Y. Velenik
Preprint
(2025).
Let $h:[0,1]\to\mathbb{R}$ be $C^2$ and such that $\sup_{[0,1]} h''<0$. For a (large) positive integer $n$, set $h_n(k) = n h(k/n)$ for any $k\in\{0,\dots,n\}$. We consider a random walk $(S_k)_{k\geq 0}$ with i.i.d. centred increments having some finite exponential moments.
We are interested in the event $\{S\geq h_n\} = \{S_k\geq h_n(k)\;\forall k\in\{0,\dots,n\}\}$.
It is well known that $P(S\geq h_n \,|\, S_0=0,\, S_n=\lceil h_n(n) \rceil) = e^{-n\int_0^1 I(h'(s)) \,ds + o(n)}$, where $I$ is the Legendre-Fenchel transform of the log-moment generating function associated to the increments. We first prove that the leading correction is of order $e^{-\Theta(n^{1/3})}$. We then turn our attention to the conditional random walk measure $P^h_n = P(\cdot \,|\, S\geq h_n, S_0=0, S_n=\lceil h_n(n) \rceil)$. We prove that the one-point tails are of the form
$\mathbb{P}_n^h (S_k \geq h_n(k) + t n^{1/3} ) = e^{-\Theta(t^{3/2})}$ for all $t
Key words:
Random walk, obstacle, large deviations, Airy, Ferrari-Spohn
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