[Paper Review] Nesting statistics in the O(n) loop model on random planar maps

Gaëtan Borot, Jérémie Bouttier+1 more|arXiv (Cornell University)|May 7, 2016

Theoretical and Computational Physics4 citations

TL;DR

This paper introduces refined generating functions to track loop nesting depth in the O(n) loop model on random planar maps using analytic combinatorics. It derives functional relations for these generating functions and explicitly computes them in a bending energy model, revealing universal large deviations behavior for nesting statistics that matches predictions from Liouville quantum gravity and conformal loop ensembles.

ABSTRACT

In the O(n) loop model on random planar maps, we study the depth - in terms of the number of levels of nesting - of the loop configuration, by means of analytic combinatorics. We focus on the 'refined' generating series of pointed disks or cylinders, which keep track of the number of loops separating the marked point from the boundary (for disks), or the two boundaries (for cylinders). For the general O(n) loop model, we show that these generating series satisfy functional relations obtained by a modification of those satisfied by the unrefined generating series. In a more specific O(n) model where loops cross only triangles and have a bending energy, we explicitly compute the refined generating series. We analyse their non generic critical behavior in the dense and dilute phases, and obtain the large deviations function of the nesting distribution, which is expected to be universal. Using the framework of Liouville quantum gravity (LQG), we show that a rigorous functional KPZ relation can be applied to the multifractal spectrum of extreme nesting in the conformal loop ensemble (CLE) in the Euclidean unit disk, as obtained by Miller, Watson and Wilson, or to its natural generalisation to the Riemann sphere. It allows us to recover the large deviations results obtained for the critical O(n) random planar map models. This offers, at the refined level of large deviations theory, a rigorous check of the fundamental fact that the universal scaling limits of random planar map models as weighted by partition functions of critical statistical models are given by LQG random surfaces decorated by independent CLEs.

Motivation & Objective

• To study the depth of loop nesting in the O(n) loop model on random planar maps using refined generating functions that track the number of loops separating a marked point from the boundary.
• To derive functional relations for these refined generating functions in the general O(n) model, modifying those of the unrefined case.
• To explicitly compute the refined generating series in a specific O(n) model with bending energy and loop crossings restricted to triangles.
• To analyze the critical behavior of the model in the dense and dilute phases, focusing on non-generic criticality and large deviations of nesting statistics.
• To establish a rigorous connection between the large deviations of nesting in random planar maps and the multifractal spectrum of CLE in Liouville quantum gravity via a functional KPZ relation.

Proposed method

• Uses analytic combinatorics to define refined generating series for pointed disks and cylinders, tracking the number of nested loops between the marked point and the boundary.
• Derives modified functional equations for the refined generating series by adapting the functional relations of the unrefined case, incorporating nesting depth information.
• Applies the framework of Liouville quantum gravity (LQG) to establish a rigorous functional KPZ relation between the large deviations spectrum in the O(n) random map model and that of the conformal loop ensemble (CLE_κ).
• Performs asymptotic analysis of special functions (e.g., trigonometric and hypergeometric terms) in the critical limit as parameters approach their critical values.
• Employs the implicit function theorem in bivariate form to prove analyticity and non-degeneracy of Jacobian matrices in the critical regime, ensuring convergence of series expansions.
• Uses power series expansions with non-negative coefficients and bounds on derivatives to control convergence and analyticity of generating functions near critical points.

Experimental results

Research questions

• RQ1 How do refined generating functions that track loop nesting depth behave in the general O(n) loop model on random planar maps?
• RQ2 What is the explicit form of the refined generating series in a specific O(n) model with bending energy and triangle-restricted loop crossings?
• RQ3 What is the critical behavior of the nesting statistics in the dense and dilute phases of the O(n) model, and how does it deviate from generic universality?
• RQ4 Can the large deviations function of the nesting distribution be derived and shown to be universal across different models?
• RQ5 To what extent does the functional KPZ relation in Liouville quantum gravity reproduce the large deviations results from the critical O(n) random planar map model?

Key findings

• The refined generating series for the O(n) loop model on random planar maps satisfy modified functional relations compared to the unrefined case, incorporating nesting depth information.
• In the bending energy model with loop crossings restricted to triangles, the refined generating series are explicitly computed using analytic combinatorics and special function asymptotics.
• The model exhibits non-generic critical behavior in both the dense and dilute phases, with distinct large deviations functions for the nesting distribution.
• The large deviations function of the nesting statistics is found to be universal and matches predictions from Liouville quantum gravity and CLE_κ via a rigorous functional KPZ relation.
• The functional KPZ relation successfully maps the large deviations spectrum of the O(n) random map model to that of CLE on the Riemann sphere, confirming the universality of LQG as the scaling limit.
• The analyticity and non-degeneracy of the Jacobian in the critical regime are rigorously established using power series bounds and the implicit function theorem, ensuring the validity of the asymptotic expansions.