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We present new exact results for the complex generalized integral means spectrum (in the sense of Duplantier et al. (Commun Math Phys 359(3):823–868, 2018)) for two kinds of whole-plane Loewner evolutions driven by a Lévy process: The case of a Lévy process with continuous trajectories, which corresponds to Schramm–Loewner evolution SLE κ with a drift term in the Brownian driving function. There is no known result for its standard integral means spectrum, and we show that a natural path to access it goes through the introduction of the complex generalized integral means spectrum, which is obtained via the so-called Liouville quantum gravity. The case of symmetric Lévy processes for which we generalize results by Loutsenko and Yermolayeva (J Phys A Math Theor 47(16):165202, 2014, J Phys A Math Theor 52(43):435202, 2019). [ABSTRACT FROM AUTHOR]

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 (sp) = 1 2 [ γ 11 Θ (sp) + γ 11 Θ ] + ν Θ . For general networks, the explicit expression of configurational exponents then naturally involves 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 cases 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. [ABSTRACT FROM AUTHOR]

We consider the ab initio continuum theory of polymer chains in a theta solvent, interacting via two- and three-body forces, with a short range cutoff s0 and in continuous space dimension d(2

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 ≥ 3 , whereas the second group comprises maps with vertices of mixed valencies, and the so-called rigid case of 2 q -regular maps (q ≥ 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 , ... , 7 , and for maps with mixed valencies (2 , 3) , (2 , 4) and (3 , 4). [ABSTRACT FROM AUTHOR]

It has been shown that for f an instance of the whole-plane SLE unbounded conformal map from the unit disk D to the slit plane, the derivative moments E(|f′(z)|p) can be written in a closed form for certain values of p depending continuously on the SLE parameter κ∈(0,∞). We generalize this property to the mixed moments, E(|f′(z)|p|f(z)|q), along integrability curves in the moment plane (p,q)∈R2 depending continuously on κ, by extending the so-called Beliaev-Smirnov equation to this case. The generalization of this integrability property to the m-fold transform of f is also given. We define a novel generalized integral means spectrum, β(p,q;κ), corresponding to the singular behavior of the above mixed moments. By inversion, it allows 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 is found to take four possible forms, separated by five phase transition lines in the moment plane R2. The average generalized spectrum of the m-fold whole-plane SLE is obtained directly from the m = 1 case by a linear map acting in the moment plane. We also give a conjecture for the precise form of the universal generalized integral means spectrum. [ABSTRACT FROM AUTHOR]

We complete the mathematical analysis of the fine structure of harmonic measure on SLE curves that was initiated in Beliaev and Smirnov (Commun Math Phys 290(2):577-595, 2009), as described by the averaged integral means spectrum. For the unbounded version of whole-plane SLE as studied in Duplantier et al. (Ann Henri Poincaré 16(6):1311-1395, 2014. ) and Loutsenko and Yermolayeva (J Stat Mech P04007, 2013), 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 of Beliaev and Smirnov, 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. [ABSTRACT FROM AUTHOR]

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 Knizhnik, Polyakov and Zamolodchikov (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. [ABSTRACT FROM AUTHOR]

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 κ ) in the Euclidean unit disk, as obtained by Miller et al. (Ann Probab 44(2):1013–1052, 2016, arXiv:1401.0217), 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. [ABSTRACT FROM AUTHOR]

Karl Löwner (later known as Charles Loewner) introduced his famous differential equation in 1923 to solve the Bieberbach conjecture for series expansion coefficients of univalent analytic functions at level n = 3. His method was revived in 1999 by Oded Schramm when he introduced the Stochastic Loewner Evolution (SLE), a conformally invariant process which made it possible to prove many predictions from conformal field theory for critical planar models in statistical mechanics. The aim of this paper is to 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 'oddified' or m-fold conformal maps of whole-plane SLEs or Lévy-Loewner Evolutions. We also study the (average) integral means multifractal spectra of these unbounded whole-plane SLE curves. We prove the existence of a phase transition at a moment order p = p( κ) > 0, at which one goes from the bulk SLE average integral means spectrum, as predicted by the first author (Duplantier Phys. Rev. Lett. 84:1363-1367, ) and established by Beliaev and Smirnov (Commun Math Phys 290:577-595, ) and valid for p ≤ p( κ), to a new integral means spectrum for p ≥ p( κ), as conjectured in part by Loutsenko (J Phys A Math Gen 45(26):265001, ). The latter spectrum is, furthermore, shown to be intimately related, via the associated packing spectrum, to the radial SLE derivative exponents obtained by Lawler, Schramm and Werner (Acta Math 187(2):237-273, ), and to the local SLE tip multifractal exponents obtained from quantum gravity by the first author (Duplantier Proc. Sympos. Pure Math. 72(2):365-482, ). This is generalized to the integral means spectrum of the m-fold transform of the unbounded whole-plane SLE map. A succinct, preliminary, version of this study first appeared in Duplantier et al. (Coefficient estimates for whole-plane SLE processes, Hal-00609774, ). [ABSTRACT FROM AUTHOR]

Gaussian Multiplicative Chaos is a way to produce a measure on $${\mathbb{R}^d}$$ (or subdomain of $${\mathbb{R}^d}$$ ) of the form $${e^{\gamma X(x)} dx}$$ , where X is a log-correlated Gaussian field and $${\gamma \in [0, \sqrt{2d})}$$ is a fixed constant. A renormalization procedure is needed to make this precise, since X oscillates between −∞ and ∞ and is not a function in the usual sense. This procedure yields the zero measure when $${\gamma = \sqrt{2d}}$$ . Two methods have been proposed to produce a non-trivial measure when $${\gamma = \sqrt{2d}}$$ . The first involves taking a derivative at $${\gamma = \sqrt{2d}}$$ (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. [ABSTRACT FROM AUTHOR]

We show that when two boundary arcs of a Liouville quantum gravity random surface are conformally welded to each other (in a boundary length-preserving way) the resulting interface is a random curve called the Schramm-Loewner evolution. We also develop a theory of quantum fractal measures (consistent with the Knizhnik-Polyakov-Zamolochikov relation) and analyze their evolution under conformal weld- ing maps related to Schramm-Loewner evolution. As an application, we construct quantum length and boundary intersection measures on the Schramm-Loewner evolution curve itself. [ABSTRACT FROM AUTHOR]

Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2 π)∫∇ h( z)⋅∇ h( z) dz, and a constant 0≤ γ<2. The Liouville quantum gravity measure on D is the weak limit as ε→0 of the measures where dz is Lebesgue measure on D and h( z) denotes the mean value of h on the circle of radius ε centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the Knizhnik, Polyakov, Zamolodchikov (Mod. Phys. Lett. A, 3:819-826, ) relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of ∂D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity. [ABSTRACT FROM AUTHOR]

Abstract: We consider random conformally invariant paths in the complex plane (SLEs). Using the Coulomb gas method in conformal field theory, we rederive the mixed multifractal exponents associated with both the harmonic measure and winding (rotation or monodromy) near such critical curves, previously obtained by quantum gravity methods. The results also extend to the general cases of harmonic measure moments and winding of multiple paths in a star configuration. [Copyright &y& Elsevier]