Random Walk conditioned to stay above a non-flat floor: curvature effects

TL;DR

This work analyzes a one-dimensional random walk conditioned to stay above a concave obstacle with negative curvature, deriving a leading large deviation form $P(S\ge h_n) \approx \exp\{ -n\int_0^1 I(h'(s))\,ds \}$ and a universal subleading correction of order $e^{-Θ(n^{1/3})}$. Using a change-of-measure to an area-tilted, time-inhomogeneous random walk in an effective potential, the authors connect the large-deviation part to a tilted kernel and obtain sharp trajectorial bounds: transversal fluctuations scale as $n^{1/3}$, longitudinal correlations scale as $n^{2/3}$, and height deviations exhibit tails of the form $e^{-Θ(t^{3/2})}$, with moments $Θ(n^{r/3})$ and covariance decay $e^{-O(|k-\ell|/n^{2/3})}$ away from the endpoints. When obstacle regularity is relaxed, the universal exponents can change; in the Gaussian case with obstacle $h(x)=1-|x|^p$, the typical height scales as $n^{α_p}$ with $α_p=(p-1)/(2p-1)$ and height-tails as $e^{-c\lambda^{(2p-1)/p}}$, illustrating non-universal behavior under weaker curvature. The Gaussian special case further yields explicit scaling predictions and rigorous bounds, bridging curvature-driven constraints with Ferrari–Spohn-type diffusion universality and informing interface phenomena in spin systems with concave walls. The methods provide a robust framework for curvature-influenced large deviations and may extend to higher-dimensional interfaces and mesoscopic obstacles.

Abstract

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 $n$.

Figures (6)

• Figure 1: The graph of $p\mapsto \alpha_p = \frac{p-1}{2p-1}$.
• Figure 2: Construction in the proof of Lemma \ref{['lem:high_exc_condUB']}. The path must remain in the shaded region and enter the narrow tube through the openings at each extremity. The endpoints must stay below the dotted lines.
• Figure 3: The construction in the proof of Lemma \ref{['lem:dev_proba_UB']}. When the event $B$ occurs, the trajectory visits both shaded rectangles. The leftmost and rightmost points at which this occurs are denoted $s$ and $t$ respectively.
• Figure 4: The construction in the proof of Lemma \ref{['lem:covariances_UB']}. The two independent paths are constrained to lie inside the shaded area. the goal is to prove that there is a reasonable probability that they intersect inside the dark shaded region.
• Figure 5: The construction in Section \ref{['subsec:Gaussian:LowerBoundHeightdeviation']}. Left: the truncated obstacle $h_n^-$ (green). Right: zoom near the plateau. The trajectory in red is the original process, conditioned to stay above $h_n$ (black); the trajectory in blue is the random walk bridge that is stochastically dominated by the original process.

Theorems & Definitions (48)

• Theorem 1.1
• proof
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• proof
• Theorem 1.3
• proof
• Theorem 1.4
• proof