The wired minimal spanning forest on the Poisson-weighted infinite tree

2022/07/19 by Nachmias, Asaf, Tang, Pengfei · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2207.09305

Abstract

We study the spectral and diffusive properties of the wired minimal spanning forest (WMSF) on the Poisson-weighted infinite tree (PWIT). Let M be the tree containing the root in the WMSF on the PWIT and (Y_n)_n≥0 be a simple random walk on M starting from the root. We show that almost surely M has ℙ[Y_2n=Y₀]=n^-3/4+o(1) and dist(Y₀,Y_n)=n^1/4+o(1) with high probability. That is, the spectral dimension of M is (3)/(2) and its typical displacement exponent is (1)/(4), almost surely. These confirm Addario-Berry's predictions in arXiv:1301.1667.

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