Convergence of scaled asymptotically-free self-interacting random walks to Brownian motion perturbed at extrema

Abstract: We consider a family of one-dimensional self interacting walks whose dynamics characterized by a monotone weight function $w$ on $\mathbb{N}\cup {0}$. The weight function takes the form $w(n) = (1 + 2^p Bn^{-p} + O(n^{-1-\kappa}))^{-1}$, for some $B \in \mathbb{R} $, $\kappa>0$ and $p\in (0,1]$. Our main model parameter is $p$, and for $p\in (0,1/2]$ we show the convergence of the SIRW to Brownian motion perturbed at extrema under the diffusive scaling. This completes the functional limit theorem in [8] for the asymptotically free case and extends the result to the full parameter range $(0,1]$. Our method depends on the generalized Ray-Knight theorems ([T96], [KMP23]) for the rescaled local times of this walk. The directed edge local times, described by the branching-like processes, are used to analyze the total drift experienced by the walker.