3 research outputs found

    Comparison of metric spectral gaps

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    Let A=(aij)∈Mn(R)A=(a_{ij})\in M_n(\R) be an nn by nn symmetric stochastic matrix. For p∈[1,∞)p\in [1,\infty) and a metric space (X,dX)(X,d_X), let Ξ³(A,dXp)\gamma(A,d_X^p) be the infimum over those γ∈(0,∞]\gamma\in (0,\infty] for which every x1,...,xn∈Xx_1,...,x_n\in X satisfy 1n2βˆ‘i=1nβˆ‘j=1ndX(xi,xj)p≀γnβˆ‘i=1nβˆ‘j=1naijdX(xi,xj)p. \frac{1}{n^2} \sum_{i=1}^n\sum_{j=1}^n d_X(x_i,x_j)^p\le \frac{\gamma}{n}\sum_{i=1}^n\sum_{j=1}^n a_{ij} d_X(x_i,x_j)^p. Thus Ξ³(A,dXp)\gamma(A,d_X^p) measures the magnitude of the {\em nonlinear spectral gap} of the matrix AA with respect to the kernel dXp:XΓ—Xβ†’[0,∞)d_X^p:X\times X\to [0,\infty). We study pairs of metric spaces (X,dX)(X,d_X) and (Y,dY)(Y,d_Y) for which there exists Ξ¨:(0,∞)β†’(0,∞)\Psi:(0,\infty)\to (0,\infty) such that Ξ³(A,dXp)≀Ψ(Ξ³(A,dYp))\gamma(A,d_X^p)\le \Psi(\gamma(A,d_Y^p)) for every symmetric stochastic A∈Mn(R)A\in M_n(\R) with Ξ³(A,dYp)<∞\gamma(A,d_Y^p)<\infty. When Ξ¨\Psi 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 n∈Nn\in \N and p∈(2,∞)p\in (2,\infty) then for every f1,...,fn∈Lpf_1,...,f_n\in L_p there exist x1,...,xn∈L2x_1,...,x_n\in L_2 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 βˆ‘i=1nβˆ‘j=1nβˆ₯xiβˆ’xjβˆ₯22=βˆ‘i=1nβˆ‘j=1nβˆ₯fiβˆ’fjβˆ₯p2. \sum_{i=1}^n\sum_{j=1}^n \|x_i-x_j\|_2^2=\sum_{i=1}^n\sum_{j=1}^n \|f_i-f_j\|_p^2. This statement is impossible for p∈[1,2)p\in [1,2), and the asymptotic dependence on pp in \eqref{eq:p factor} is sharp. We also obtain the best known lower bound on the LpL_p 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
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