Journal: Annales de l'Institut Henri Poincaré D. Combinatorics, Physics and their Interactions

Abbreviation

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

Publisher

European Mathematical Society

Journal Volumes

ISSN

2308-5827
2308-5835

Description

Filters

2020 - 20232
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Publications 1 - 2 of 2
  • The perimeter cascade in critical Boltzmann quadrangulations decorated by an O(n) loop model
    Item type: Journal Article
    Chen, Linxiao; Curien, Nicolas; Maillard, Pascal (2020)
    Annales de l'Institut Henri Poincaré D. Combinatorics, Physics and their Interactions
    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.
  • Tropical Monte Carlo quadrature for Feynman integrals
    Item type: Journal Article
    Borinsky, Michael (2023)
    Annales de l'Institut Henri Poincaré D. Combinatorics, Physics and their Interactions
    We introduce a new method to evaluate algebraic integrals over the simplex numerically. This new approach employs techniques from tropical geometry and exceeds the capabilities of existing numerical methods by an order of magnitude. The method can be improved further by exploiting the geometric structure of the underlying integrand. As an illustration of this, we give a specialized integration algorithm for a class of integrands that exhibit the form of a generalized permutahedron. This class includes integrands for scattering amplitudes and parametric Feynman integrals with tame kinematics. A proof-of-concept implementation is provided with which Feynman integrals up to loop order 17 can be evaluated.
Publications 1 - 2 of 2