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 $\Omega(\log^2 n)$, and the local total variation distance exhibits non-concentrated decay, formally preventing a sharp cutoff phenomenon.
We consider the upper tail large deviations of subgraph counts for irregular graphs H$$ \mathrm{H} $$ in 𝔾(n,p) , the sparse Erdős‐Rényi graph on n$$ n $$ vertices with edge connectivity probability p∈(0,1)$$ p\in \left(0,1\right) $$ . For n−1/Δ≪p≪1$$ {n}^{-1/\Delta}\ll p\ll 1 $$ , where Δ$$ \Delta $$ is the maximum de...
Anirban Basak, Shaibal Karmakar· Random Structures & Algo...· 4 citations· ⚡3
We consider the adjacency matrix of a uniformly random simple $d$-regular graph on $N$ vertices. For every fixed $d\geq3$ and every fixed bulk energy, we prove that the rescaled eigenvalue point process converges to the $\mathrm{Sine}_1$ process with intensity $1/\pi$. We also establish universality of consecutive gaps...
In Markov chain Monte Carlo sampling, light-tailed target distributions present something of a poisoned chalice: their light tails offer good confinement, and tend to imply good mixing properties for natural continuous-time dynamics, but the steepness of their tail decay means that they often fall outside of the scope...
We study continuous-time supercritical branching random walks in space-inhomogeneous random environment on the Hamming cube $\{0,1,\dots,b-1\}^d$, where the reproduction laws at each site are i.i.d. sampled. This serves as an idealized model for RNA sequence evolution with mutation--selection balance. The reproduction...
We introduce general methods to derive lower tail large deviation principles, for a variety of nonlinear problems of combinatorial interest. These methods are especially effective in"critical"regimes, where the nonlinearities are not sparse enough for Poissonian lower tails (e.g., Janson's inequality does not provide a...
Repeated averaging on graphs is a natural prototype of local stochastic exchange, modeling consensus formation, information spreading, and wealth redistribution. Its convergence has attracted longstanding interest, from Bourgain's complete-graph $L^1$ cutoff question in the 80s (answered in recent years by Chatterjee,...
Dong Yao, Ling-Fu Zhang· 0 citations
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