Let A=(aijβ)βMnβ(R) be an n by n symmetric stochastic matrix. For
pβ[1,β) and a metric space (X,dXβ), let Ξ³(A,dXpβ) be the
infimum over those Ξ³β(0,β] for which every x1β,...,xnββX
satisfy n21βi=1βnβj=1βnβdXβ(xiβ,xjβ)pβ€nΞ³βi=1βnβj=1βnβaijβdXβ(xiβ,xjβ)p.
Thus Ξ³(A,dXpβ) measures the magnitude of the {\em nonlinear spectral
gap} of the matrix A with respect to the kernel dXpβ:XΓXβ[0,β). We study pairs of metric spaces (X,dXβ) and (Y,dYβ) for which
there exists Ξ¨:(0,β)β(0,β) such that Ξ³(A,dXpβ)β€Ξ¨(Ξ³(A,dYpβ)) for every symmetric stochastic AβMnβ(R) with
Ξ³(A,dYpβ)<β. When Ξ¨ is linear a complete geometric
characterization is obtained.
Our estimates on nonlinear spectral gaps yield new embeddability results as
well as new nonembeddability results. For example, it is shown that if nβN and pβ(2,β) then for every f1β,...,fnββLpβ there exist
x1β,...,xnββL2β such that {equation}\label{eq:p factor} \forall\, i,j\in
\{1,...,n\},\quad \|x_i-x_j\|_2\lesssim p\|f_i-f_j\|_p, {equation} and i=1βnβj=1βnββ₯xiββxjββ₯22β=i=1βnβj=1βnββ₯fiββfjββ₯p2β.
This statement is impossible for pβ[1,2), and the asymptotic dependence
on p in \eqref{eq:p factor} is sharp. We also obtain the best known lower
bound on the Lpβ distortion of Ramanujan graphs, improving over the work of
Matou\v{s}ek. Links to Bourgain--Milman--Wolfson type and a conjectural
nonlinear Maurey--Pisier theorem are studied.Comment: Clarifying remarks added, definition of p(n,d) modified, typos fixed,
references adde