High-dimensional random walks (Mon, Feb 27)

Trickle Down Theorem

Proof:

Corollary

Proof:

Random walks

Define \(\e_k \seteq \min_{\sigma \in X^{(k)}} \e(\pi^{(1)}_{\sigma})\) as the minimum spectral gap over all \(1\)-skeleta of links of \(k\)-dimensional faces.

Then for every \(0 \leq k \leq d-1\), and \(f \in \ell_2(\pi^{(k)})\), it holds that

\[\begin{equation}\label{eq:al-thm} \langle f, L_k^{\wedge} f\rangle_{\pi^{(k)}} \geq \e_{k-1} \langle f, L_k^{\triangledown} f\rangle_{\pi^{(k)}}\,. \end{equation}\]

Corollary

Proof of \eqref{eq:al-thm}: