# SIAM Chapter Seminar: On the weighted asymptotic number of simplicial spanning trees in random $$k$$-regular simplicial complexes

#### 24.11.2021, 4:30 pm - 5:00 pm  –  Golm, Haus 9, Raum 2.22 and Zoom

Lior Tenenbaum

Abstract: In 1981, Mckay proved an asymptotic result regarding the number of spanning trees in random $$k$$-regular graphs. In this talk we will discuss an analogous result for random high dimensional $$k$$-regular simplicial complexes, showing that the weighted number of simplicial spanning trees in such complexes converge asymptotically almost surely to an explicit constant $$\zeta_{d,k}$$, when $$n$$ tends to infinity provided that $$k\geq 2d^2+2d\sqrt{d^2-1}$$. A key ingredient in the proof is the local convergence of such random complexes to the $$d$$-dimensional, $$k$$-regular arboreal complex, which allows us to generalize Mckay's result regarding the Kesten-Mckay distribution.

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