Random Walk conditioned to stay above a non-flat floor: curvature effects
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866918198754934784 |
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| author | Ott, Sébastien Velenik, Yvan |
| author_facet | Ott, Sébastien Velenik, Yvan |
| contents | 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 $e^{-Θ(n^{1/3})}$. 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 $\mathbb{P}_n^h (S_k \geq h_n(k) + t n^{1/3} ) = e^{-Θ(t^{3/2})}$ for all $t<n^β$ for any $β\in (0,1/6)$. Moreover, we prove that, for any $r\geq 1$, $E_n^h((S_k-h_n(k))^r) = Θ(n^{r/3})$ and $\mathrm{Var}_{P_n^h}(S_k) = Θ(n^{2/3})$, for all $k$ far enough from $0$ and $n$. In addition, we show that $\mathrm{Cov}_{P_n^h}(S_k,S_\ell) \leq e^{-O(|\ell-k|/n^{2/3})}$ for all $k,\ell$ not too close to $0$ and $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09280 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random Walk conditioned to stay above a non-flat floor: curvature effects Ott, Sébastien Velenik, Yvan Probability Statistical Mechanics Mathematical Physics 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 $e^{-Θ(n^{1/3})}$. 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 $\mathbb{P}_n^h (S_k \geq h_n(k) + t n^{1/3} ) = e^{-Θ(t^{3/2})}$ for all $t<n^β$ for any $β\in (0,1/6)$. Moreover, we prove that, for any $r\geq 1$, $E_n^h((S_k-h_n(k))^r) = Θ(n^{r/3})$ and $\mathrm{Var}_{P_n^h}(S_k) = Θ(n^{2/3})$, for all $k$ far enough from $0$ and $n$. In addition, we show that $\mathrm{Cov}_{P_n^h}(S_k,S_\ell) \leq e^{-O(|\ell-k|/n^{2/3})}$ for all $k,\ell$ not too close to $0$ and $n$. |
| title | Random Walk conditioned to stay above a non-flat floor: curvature effects |
| topic | Probability Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2511.09280 |