Spectral gap, isoperimetry and concentration on trees

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We consider the nearest-neighbor model on the finite tree T with generator L. We obtain a twosided estimate of the spectral gap by factor 2. We also identify explicitly the Lipschitzian norm of the operator (–L)−1 in propriate functional space. This leads to the identification of the best constant in the generalized Cheeger isoperimetric inequality on the tree, and to transportation-information inequalities.