The perimeter cascade in critical Boltzmann quadrangulations decorated by an O(n) loop model
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Date
2020
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Journal Article
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Abstract
We study the branching tree of the perimeters of the nested loops in the non-generic critical O(n) model on random quadrangulations. We prove that after renormalization it converges towards an explicit continuous multiplicative cascade whose offspring distribution (x(i))(i >= 1) is related to the jumps of a spectrally positive alpha-stable Levy process with alpha = 3/2 +/- 1/pi arccos(n/2) and for which we have the surprisingly simple and explicit transform E[Sigma(i >= 1)(x(i))(theta)] = sin(pi(2 - alpha))/sin(pi(theta - alpha)), for theta is an element of (alpha, alpha + 1). An important ingredient in the proof is a new formula of independent interest on first moments of additive functionals of the jumps of a left-continuous random walk stopped at a hitting time. We also identify the scaling limit of the volume of the critical O(n)-decorated quadrangulation using the Malthusian martingale associated to the continuous multiplicative cascade.
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published
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7 (4)
Pages / Article No.
535 - 584
Publisher
European Mathematical Society
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Subject
O(n) loop model; random planar map; multiplicative cascade; invariance principle; stable Levy process