A self-repelling walk on the integers avoids places it has visited frequently — each return to a site increases the local repulsion, biasing future steps away. The strength of repulsion decays polynomially with the visit count, governed by a parameter alpha that controls how quickly memory fades. Generalized Polya urns provide the analytical framework: each site's accumulated visits function as the urn's composition, and the walk's trajectory emerges from the urn dynamics. Extending the theory from a specific weight function to the general family w(n)^{-1} = n^alpha(1 + 2Bn^{-1} + O(n^{-2})) opens the path to proving scaling limits — characterizing the walk's long-time diffusive behavior.
The mathematical precision of the extension matters because the subleading correction term B controls the walk's large-scale behavior. The leading-order alpha determines whether the walk is diffusive, superdiffusive, or ballistic, but B determines the prefactors and fluctuation structure within each regime. Getting the asymptotics right to second order is not pedantry — it is the difference between knowing the qualitative behavior and predicting the quantitative one.
The principle: in systems with memory, the dominant effect determines the regime, but the first correction determines the behavior within that regime. Subleading terms carry the information that distinguishes one self-interacting system from another.
(arXiv:2603.03622)