Word maps and spectra of random graph lifts

Word maps and spectra of random graph lifts

We study here the spectra of random lifts of graphs. Let G be a finite connected graph, and let the infinite tree T be its universal cover space. If λ1 and ρ are the spectral radii of G and T respectively, then, as shown by Friedman [Fri03], in almost every n-lift H of G, all “new” eigenvalues of H are ≤ O ( λ 1/2 1 ρ 1/2 ) . Here we improve this bound to O ( λ 1/3 1 ρ 2/3 ) . It is conjectured...

Random Graph Lifts

Let G be any connected graph, and let λ1 and ρ denote the spectral radius of G and the universal cover of G, respectively. In [LP09], Linial and Puder have shown that almost every n-lift of G has all of its new eigenvalues bounded by O(λ 1/3 1 ρ 2/3). Friedman had conjectured that this bound can be improved to ρ+ on(1) (e.g., see [Fri03, HLW06]). In [LP09], Linial and Puder have formulated two ...

Random Lifts of Graphs III :

Hamiltonian flows, cotangent lifts, and momentum maps

Let (M,ω) and (N, η) be symplectic manifolds. A symplectomorphism F : M → N is a diffeomorphism such that ω = F ∗η. Recall that for x ∈ M and v1, v2 ∈ TxM , (F η)x(v1, v2) = ηF (x)((TxF )v1, (TxF )v2); TxF : TxM → TF (x)N . (A tangent vector at x ∈ M is pushed forward to a tangent vector at F (x) ∈ N , while a differential 2-form on N is pulled back to a differential 2-form on M .) In these not...

Asymptotic Lifts of Positive Linear Maps

We show that the notion of asymptotic lift generalizes naturally to normal positive maps φ : M → M acting on von Neumann algebras M . We focus on cases in which the domain of the asymptotic lift can be embedded as an operator subsystem M∞ ⊆ M , and characterize when M∞ is a Jordan subalgebra of M in terms of the asymptotic multiplicative properties of φ.