arXiv subjects

Bertrand Duplantier

Publications and source records attributed to Bertrand Duplantier.

At least 19 recordsLinked to original sources

arXiv2026

Liouville Quantum Duality and Random Planar Maps II

This is Part II of our project on block-weighted planar maps and Liouville quantum duality. Focusing on the scaling properties at the dual critical point, we derive the conditional distribution of the root block size given the total size, as well as, conversely, the distribution of the total size for a fixed root block size. We show that these laws are in perfect agreement with the results of Liouville quantum gravity (LQG), obtained by modifying the standard Liouville random measure with additional atomic contributions representing localized quantum areas. The ratio of dual and direct partition functions with punctures is shown to be universal, its explicit LQG expression exactly matching its combinatorial analogue. We also investigate the block distance profile for doubly rooted maps, which is here rigorously related to the distance profile of maps consisting of a single block. Finally, we analyze the multifractal properties of the usual and dual Liouville measures, predicting the associated spectra, from both quantum and Euclidean perpectives. We illustrate our results through specific realizations of block-weighted planar maps, i.e., quadrangulations decomposed into simple blocks, tree-like structures formed by attaching quartic maps, and bicubic maps decomposed into 3-connected blocks. For each model, we give the single non-universal constant which uniquely determines the strength of the corresponding atomic Liouville measure.

math-ph↗

arXiv2026

Integral Means Spectrum for the Random Riemann Zeta Function

We study the integral means spectrum associated with the analytic function whose derivative is the so-called randomized Riemann zeta-function, introduced some time ago by Bagchi. The randomized $ζ$-function, $ζ_{\mathrm{rand}}(σ+ih)$, is known to represent the asymptotic statistical behaviour of the random vertical shifts of the actual $ζ$-function in the critical strip, $1/2 <σ\leq 1, h\in \mathbb R$, and appears in a number of recent works on the asymptotic behavior of the moments and maxima of the $ζ$-function on short intervals along the critical axis $σ=1/2$. Using probability and basic analytic number theory, we show that the complex integral means spectrum of the primitive of $ζ_{\mathrm{rand}}$ is almost surely of the form conjectured 30 years ago by Kraetzer, for the so-called universal integral means spectrum of univalent functions in the disc. The Riemann $ζ$-function and its random version have recently been rigorously related to the so-called Gaussian multiplicative chaos (GMC), initiated by Kahane 40 years ago. In the case of the holomorphic multiplicative chaos on the unit disc -- an important stochastic object closely related to Liouville quantum gravity on the unit circle -- we prove that the integral means spectrum of the primitive is almost surely also of the same Kraetzer form. However, we establish that neither the primitive of the random function $ζ_{\mathrm{rand}}$, nor that of the holomorphic GMC are injective. Building on earlier work by one of the authors and Webb on the convergence of Riemann $ζ$-function on the critical line to a holomorphic GMC distribution, we finally provide an alternative derivation of the integral means spectrum for the random Riemann $ζ$-function.

math.CV↗

arXiv2025

Liouville Quantum Duality and Random Planar Maps

We consider models of block-weighted random planar maps in which possibly decorated maps are decomposed canonically into blocks, each block receiving the weight $u$. These maps present a transition at some critical value $u=u_{cr}$ above which the maps degenerate into Brownian trees. We show that the enumerative properties and critical exponents of the maps at $u=u_{cr}$ and those for $u<u_{cr}$ are connected by duality relations which are precisely those expected in the context of the Liouville quantum gravity description of random surfaces. We illustrate this result by various instances of block-weighted maps: random planar quadrangulations decomposed into simple blocks, Hamiltonian cycles on cubic or bicubic planar maps decomposed into irreducible blocks, and meandric systems.

math-ph↗

arXiv2023

Nesting statistics in the O(n) loop model on random planar maps

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.

math-ph↗

arXiv2023

Hamiltonian cycles on bicolored random planar maps

We study the statistics of Hamiltonian cycles on various families of bicolored random planar maps (with the spherical topology). These families fall into two groups corresponding to two distinct universality classes with respective central charges $c=-1$ and $c=-2$. The first group includes generic $p$-regular maps with vertices of fixed valency $p\geq 3$, whereas the second group comprises maps with vertices of mixed valencies, and the so-called rigid case of $2q$-regular maps ($q\geq 2$) for which, at each vertex, the unvisited edges are equally distributed on both sides of the cycle. We predict for each class its universal configuration exponent $γ$, as well as a new universal critical exponent $ν$ characterizing the number of long-distance contacts along the Hamiltonian cycle. These exponents are theoretically obtained by using the Knizhnik, Polyakov and Zamolodchikov (KPZ) relations, with the appropriate values of the central charge, applied, in the case of $ν$, to the corresponding critical exponent on regular (hexagonal or square) lattices. These predictions are numerically confirmed by analyzing exact enumeration results for $p$-regular maps with $p=3,4,\ldots,7$, and for maps with mixed valencies $(2,3)$, $(2,4)$ and $(3,4)$.

math-ph↗

arXiv2023

Complex Generalized Integral Means Spectrum of Drifted Whole-Plane SLE and LLE

We present new results for the complex generalized integral means spectrum for two kinds of whole-plane Loewner evolutions driven by Lévy processes: - Lévy processes with continuous trajectories, which correspond to Schramm-Loewner evolutions (SLE) with a drift term in the Brownian driving function. A natural path to access the standard integral means spectrum in the presence of drift goes through the introduction of the complex generalized integral means spectrum, which is obtained via the so-called Liouville quantum gravity. -Symmetric Lévy processes for which we generalize recent results by Loutsenko and Yermolayeva.

math-ph↗

arXiv2022

Exponents for Hamiltonian paths on random bicubic maps and KPZ

We evaluate the configuration exponents of various ensembles of Hamiltonian paths drawn on random planar bicubic maps. These exponents are estimated from the extrapolations of exact enumeration results for finite sizes and compared with their theoretical predictions based on the KPZ relations, as applied to their regular counterpart on the honeycomb lattice. We show that a naive use of these relations does not reproduce the measured exponents but that a simple modification in their application may possibly correct the observed discrepancy. We show that a similar modification is required to reproduce via the KPZ formulas some exactly known exponents for the problem of unweighted fully packed loops on random planar bicubic maps.

math-ph↗

arXiv2020

Liouville quantum gravity as a mating of trees

There is a simple way to "glue together" a coupled pair of continuum random trees (CRTs) to produce a topological sphere. The sphere comes equipped with a measure and a space-filling curve (which describes the "interface" between the trees). We present an explicit and canonical way to embed the sphere in ${\mathbf C} \cup \{ \infty \}$. In this embedding, the measure is Liouville quantum gravity (LQG) with parameter $γ\in (0,2)$, and the curve is space-filling SLE$_{κ'}$ with $κ' = 16/γ^2$. Achieving this requires us to develop an extensive suite of tools for working with LQG surfaces. We explain how to conformally weld so-called "quantum wedges" to obtain new quantum wedges of different weights. We construct finite-volume quantum disks and spheres of various types, and give a Poissonian description of the set of quantum disks cut off by a boundary-intersecting SLE$_κ(ρ)$ process with $κ\in (0,4)$. We also establish a Lévy tree description of the set of quantum disks to the left (or right) of an SLE$_{κ'}$ with $κ' \in (4,8)$. We show that given two such trees, sampled independently, there is a.s. a canonical way to "zip them together" and recover the SLE$_{κ'}$. The law of the CRT pair we study was shown in an earlier paper to be the scaling limit of the discrete tree/dual-tree pair associated to an FK-decorated random planar map (RPM). Together, these results imply that FK-decorated RPM scales to CLE-decorated LQG in a certain "tree structure" topology.

math.PR↗

arXiv2020

Statistical Mechanics of Confined Polymer Networks

We show how the theory of the critical behaviour of $d$-dimensional polymer networks of arbitrary topology can be generalized to the case of networks confined by hyperplanes. This in particular encompasses the case of a single polymer chain in a bridge configuration. We further define multi-bridge networks, where several vertices are in local bridge configurations. We consider all cases of ordinary, mixed and special surface transitions, and polymer chains made of self-avoiding walks, or of mutually-avoiding walks, or at the tricritical $Θ$-point. In the $Θ$-point case, generalising the good-solvent case, we relate the critical exponent for simple bridges, $γ_b^Θ$, to that of terminally-attached arches, $γ_{11}^Θ,$ and to the correlation length exponent $ν^Θ.$ We find $γ_b^Θ = γ_{11}^Θ+ν^Θ.$ In the case of the special transition, we find $γ_b^Θ({\rm sp}) = \frac{1}{2}[γ_{11}^Θ({\rm sp})+γ_{11}^Θ]+ν^Θ.$ For general networks, the explicit expression of configurational exponents then naturally involve bulk and surface exponents for multiple random paths. In two-dimensions, we describe their Euclidean exponents from a unified perspective, using Schramm-Loewner Evolution (SLE) in Liouville quantum gravity (LQG), and the so-called KPZ relation between Euclidean and LQG scaling dimensions. This is done in the case of ordinary, mixed and special surface transitions, and of the $Θ$-point. We provide compelling numerical evidence for some of these results both in two- and three-dimensions.

math-ph↗

arXiv2019

New scaling laws for self-avoiding walks: bridges and worms

We show how the theory of the critical behaviour of $d$-dimensional polymer networks gives a scaling relation for self-avoiding {\em bridges} that relates the critical exponent for bridges $γ_b$ to that of terminally-attached self-avoiding arches, $γ_{1,1},$ and the {correlation} length exponent $ν.$ We find $γ_b = γ_{1,1}+ν.$ We provide compelling numerical evidence for this result in both two- and three-dimensions. Another subset of SAWs, called {\em worms}, are defined as the subset of SAWs whose origin and end-point have the same $x$-coordinate. We give a scaling relation for the corresponding critical exponent $γ_w,$ which is $γ_w=γ-ν.$ This too is supported by enumerative results in the two-dimensional case.

cond-mat.stat-mech↗

arXiv2017

Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE

We consider the whole-plane SLE conformal map f from the unit disk to the slit plane, and show that its mixed moments, involving a power p of the derivative modulus |f'| and a power q of the map |f| itself, have closed forms along some integrability curves in the (p,q) moment plane, which depend continuously on the SLE parameter kappa. The generalization of this integrability property to the m-fold transform of f is also given. We define a generalized integral means spectrum corresponding to the singular behavior of the mixed moments above. By inversion, it allows for a unified description of the unbounded interior and bounded exterior versions of whole-plane SLE, and of their m-fold generalizations. The average generalized spectrum of whole-plane SLE takes four possible forms, separated by five phase transition lines in the moment plane, whereas the average generalized spectrum of the m-fold whole-plane SLE is directly obtained from a linear map acting in that plane. We also conjecture the form of the universal generalized integral means spectrum.

math-ph↗

arXiv2017

Integral means spectrum of whole-plane SLE

We complete the mathematical analysis of the fine structure of harmonic measure on SLE curves that was initiated by Beliaev and Smirnov, as described by the averaged integral means spectrum. For the unbounded version of whole-plane SLE as studied by Duplantier, Nguyen, Nguyen and Zinsmeister, and Loutsenko and Yermolayeva, a phase transition has been shown to occur for high enough moments from the bulk spectrum towards a novel spectrum related to the point at infinity. For the bounded version of whole-plane SLE studied here, a similar transition phenomenon, now associated with the SLE origin, is proved to exist for low enough moments, but we show that it is superseded by the earlier occurrence of the transition to the SLE tip spectrum.

math-ph↗

arXiv2016

A Fast Direct Sampling Algorithm for Equilateral Closed Polygons

Sampling equilateral closed polygons is of interest in the statistical study of ring polymers. Over the past 30 years, previous authors have proposed a variety of simple Markov chain algorithms (but have not been able to show that they converge to the correct probability distribution) and complicated direct samplers (which require extended-precision arithmetic to evaluate numerically unstable polynomials). We present a simple direct sampler which is fast and numerically stable, and analyze its runtime using a new formula for the volume of equilateral polygon space as a Dirichlet-type integral.

cond-mat.stat-mech↗

arXiv2014

Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.

math.PR↗

arXiv2014

Log-correlated Gaussian fields: an overview

We survey the properties of the log-correlated Gaussian field (LGF), which is a centered Gaussian random distribution (generalized function) $h$ on $\mathbb R^d$, defined up to a global additive constant. Its law is determined by the covariance formula $$\mathrm{Cov}\bigl[ (h, ϕ_1), (h, ϕ_2) \bigr] = \int_{\mathbb R^d \times \mathbb R^d} -\log|y-z| ϕ_1(y) ϕ_2(z)dydz$$ which holds for mean-zero test functions $ϕ_1, ϕ_2$. The LGF belongs to the larger family of fractional Gaussian fields obtained by applying fractional powers of the Laplacian to a white noise $W$ on $\mathbb R^d$. It takes the form $h = (-Δ)^{-d/4} W$. By comparison, the Gaussian free field (GFF) takes the form $(-Δ)^{-1/2} W$ in any dimension. The LGFs with $d \in \{2,1\}$ coincide with the 2D GFF and its restriction to a line. These objects arise in the study of conformal field theory and SLE, random surfaces, random matrices, Liouville quantum gravity, and (when $d=1$) finance. Higher dimensional LGFs appear in models of turbulence and early-universe cosmology. LGFs are closely related to cascade models and Gaussian branching random walks. We review LGF approximation schemes, restriction properties, Markov properties, conformal symmetries, and multiplicative chaos applications.

math.PR↗

arXiv2014

The Coefficient Problem and Multifractality of Whole-Plane SLE and LLE

We revisit the Bieberbach conjecture in the framework of SLE processes and, more generally, Lévy processes. The study of their unbounded whole-plane versions leads to a discrete series of exact results for the expectations of coefficients and their variances, and, more generally, for the derivative moments of some prescribed order p. These results are generalized to the m-fold conformal maps of whole-plane SLEs or Lévy-Loewner Evolutions (LLEs). We also study the (averaged) integral means multifractal spectra of these unbounded whole-plane SLE curves. We prove the existence of a phase transition at a certain moment order, at which one goes from the bulk SLE expected integral means spectrum, as established by Beliaev and Smirnov, to a new integral means spectrum. The latter is furthermore shown to be intimately related, via the associated packing spectrum, to radial SLE derivative exponents, and to local SLE tip multifractal exponents obtained from quantum gravity. This is generalized to the integral means spectrum of the m-fold transform of the unbounded whole-plane SLE map.

math-ph↗

arXiv2013

Renormalization of Critical Gaussian Multiplicative Chaos and KPZ formula

Gaussian Multiplicative Chaos is a way to produce a measure on $\R^d$ (or subdomain of $\R^d$) of the form $e^{γX(x)} dx$, where $X$ is a log-correlated Gaussian field and $γ\in [0,\sqrt{2d})$ is a fixed constant. A renormalization procedure is needed to make this precise, since $X$ oscillates between $-\infty$ and $\infty$ and is not a function in the usual sense. This procedure yields the zero measure when $γ=\sqrt{2d}$. Two methods have been proposed to produce a non-trivial measure when $γ=\sqrt{2d}$. The first involves taking a derivative at $γ=\sqrt{2d}$ (and was studied in an earlier paper by the current authors), while the second involves a modified renormalization scheme. We show here that the two constructions are equivalent and use this fact to deduce several quantitative properties of the random measure. In particular, we complete the study of the moments of the derivative multiplicative chaos, which allows us to establish the KPZ formula at criticality. The case of two-dimensional (massless or massive) Gaussian free fields is also covered.

math.PR↗

arXiv2011

The Hausdorff Dimension of Two-Dimensional Quantum Gravity

We argue that the Hausdorff dimension D of a quantum gravity random surface is always D=4, irrespective of the conformal central charge c, c between -2 and 1, of a critical statistical model possibly borne by it. The Knizhnik-Polyakov-Zamolodchikov (KPZ) relation allows one to determine the dimension D from the exact Hausdorff dimension of random maps with large faces, recently rigorously studied by Le Gall and Miermont, and reformulated by Borot, Bouttier and Guitter as a loop model on a random quadrangulation. This contradicts several earlier conjectured formulae for D in terms of c.

math-ph↗