144 research outputs found
Eigenvalue confinement and spectral gap for random simplicial complexes
We consider the adjacency operator of the Linial-Meshulam model for random
simplicial complexes on vertices, where each -cell is added
independently with probability to the complete -skeleton. Under the
assumption , we prove that the spectral gap between the
smallest eigenvalues and the remaining
eigenvalues is with high probability.
This estimate follows from a more general result on eigenvalue confinement. In
addition, we prove that the global distribution of the eigenvalues is
asymptotically given by the semicircle law. The main ingredient of the proof is
a F\"uredi-Koml\'os-type argument for random simplicial complexes, which may be
regarded as sparse random matrix models with dependent entries.Comment: 29 pages, 6 figure
The need for speed : Maximizing random walks speed on fixed environments
We study nearest neighbor random walks on fixed environments of
composed of two point types : and for . We show
that for every environment with density of drifts bounded by we
have , where
is a random walk on the environment. In addition up to some integer effect the
environment which gives the best speed is given by equally spaced drifts
- β¦