expander. In this paper we show that the spectral expansion of the links, including the spectral expansion of the $1$-dimensional skeleton of the complex, imply the spectral expansion of $G_i$ for [*all*]{} $0 < i < d$.

Parzanchevski and Rosenthal in [@PR12] study a different notion of high order random walk. The high order random walk of [@PR12] is designed to expose the topological properties of the complex. Parzanchevski and Rosenthal define a variant of the stationary distribution of the random walk and show its relation to the spectrum of the high order laplacian on the space that is [*orthogonal to the coboundaries*]{}. In our work, the stationary distribution of the random walks is already known. Moreover, it is already known that the convergence rate is controlled by the spectrum of the high order normalized adjacency matrices on the space that is [*orthogonal to the constant functions*]{}. We show that the expansion of the links implies the concentration of the spectrum of the high order normalized adjacency matrices, a thing that can not be deduced in any way from [@PR12].

We study the spectrum of the high order normalized adjacency matrices of the complex and derive some good bounds on 