Peripheral Traps and Lower Bounds on Mixing Times for Random Walks on Sparse Heavy-Tailed Random Intersection Graphs

Abstract

This paper analyzes mixing time lower bounds for random walks on sparse, heavy-tailed Random Intersection Graphs. In sparse feature regimes, heavy-tailed feature distributions lead to the formation of peripheral trap -- chains of overlapping low-weight feature cliques attached to high-weight hub nodes within the graph's giant component. By modeling escape trajectories from these traps as continuous limit hitting times for reflected Brownian motion, the analysis demonstrates that random walks experience logarithmic squared delays. Consequently, the mixing time is bounded below by $Ω(\log^2 n)$, and the local total variation distance exhibits non-concentrated decay, formally preventing a sharp cutoff phenomenon.