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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 $\gamma$, as well as a new universal critical exponent $\nu$ 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 $\nu$, 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 $(3,4)$., Comment: 41 pages, 19 figures, 9 tables

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 $\gamma \in (0,2)$, and the curve is space-filling SLE$_{\kappa'}$ with $\kappa' = 16/\gamma^2$. 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$_{\kappa}(\rho)$ process with $\kappa \in (0,4)$. We also establish a L\'evy tree description of the set of quantum disks to the left (or right) of an SLE$_{\kappa'}$ with $\kappa' \in (4,8)$. We show that given two such trees, sampled independently, there is a.s. a canonical way to "zip them together" and recover the SLE$_{\kappa'}$. 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., Comment: 209 pages, approximately 60 figures; revised

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., Comment: 14 pages, 1 figure; final version

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., Comment: 71 pages, 11 figures. v2: minor text and abstract edits, references added. v2: 105 pages, revised and expanded, including justification of delta analyticity

Gaussian Multiplicative Chaos is a way to produce a measure on $\R^d$ (or subdomain of $\R^d$) 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 $-\infty$ and $\infty$ 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., Comment: The new version contains the proofs for Free Fields

We revisit the Bieberbach conjecture in the framework of SLE processes and, more generally, L\'evy 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 L\'evy-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., Comment: 95 pages, 10 figures. Revised and improved version with 10 pages and 4 figures added; Section 4.2 extended with new subsections 4.2.4 and 4.2.7. The new Theorem 1.4 rigorously establishes the form of the expected integral means spectrum for the unbounded whole-plane SLE (for a certain interval of moment order p)

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.

We present a (mathematically rigorous) probabilistic and geometrical proof of the KPZ relation between scaling exponents in a Euclidean planar domain D and in Liouville quantum gravity. It uses the properly regularized quantum area measure d\mu_\gamma=\epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)}dz, where dz is Lebesgue measure on D, \gamma is a real parameter, 0\leq \gamma 2 is shown to be related to the quantum measure d\mu_{\gamma'}, \gamma' < 2, by the fundamental duality \gamma\gamma'=4., Comment: 4 pages, 1 figure

We describe in detail the history of Brownian motion, as well as the contributions of Einstein, Sutherland, Smoluchowski, Bachelier, Perrin and Langevin to its theory. The always topical importance in physics of the theory of Brownian motion is illustrated by recent biophysical experiments, where it serves, for instance, for the measurement of the pulling force on a single DNA molecule. In a second part, we stress the mathematical importance of the theory of Brownian motion, illustrated by two chosen examples. The by-now classic representation of the Newtonian potential by Brownian motion is explained in an elementary way. We conclude with the description of recent progress seen in the geometry of the planar Brownian curve. At its heart lie the concepts of conformal invariance and multifractality, associated with the potential theory of the Brownian curve itself., 107 pages, 21 figures, expanded and updated version of the article originally published in Einstein, 1905-2005, Poincar\'e Seminar 2005 (Birkh\"auser Verlag, 2006). Original version also available at http://www.birkhauser.ch/3-7643-7435-7

In these Notes, a comprehensive description of the universal fractal geometry of conformally-invariant scaling curves or interfaces, in the plane or half-plane, is given. The present approach focuses on deriving critical exponents associated with interacting random paths, by exploiting their underlying quantum gravity structure. The latter relates exponents in the plane to those on a random lattice, i.e., in a fluctuating metric, using the so-called Knizhnik, Polyakov and Zamolodchikov (KPZ) map. This is accomplished within the framework of random matrix theory and conformal field theory, with applications to geometrical critical models, like Brownian paths, self-avoiding walks, percolation, and more generally, the O(N) or Q-state Potts models and, last but not least, Schramm's Stochastic Loewner Evolution (SLE_kappa). These Notes can be considered as complementary to those by Wendelin Werner (2006 Fields Medalist!), ``Some Recent Aspects of Random Conformally Invariant Systems,'' arXiv:math.PR/0511268., Comment: 116 pages, 45 figures. Les Houches 2005 Lecture Notes. Based on the previous research survey article ``Conformal Fractal Geometry and Boundary Quantum Gravity,'' arXiv:math-ph/0303034 [in: ``Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot'' (M. L. Lapidus and M.van Frankenhuysen, eds.), Proc.Symposia Pure Math. vol. 72, Part 2, 365-482 (AMS, Providence, R.I., 2004], here augmented by introductory sections, explanatory figures and some new material. Supplementary technical appendices can be found in the foregoing reference

When the vacuum is partitioned by material boundaries with arbitrary shape, one can define the zero-point energy and the free energy of the electromagnetic waves in it: this can be done, independently of the nature of the boundaries, in the limit that they become perfect conductors, provided their curvature is finite. The first examples we consider are Casimir's original configuration of parallel plates, and the experimental situation of a sphere in front of a plate. For arbitrary geometries, we give an explicit expression for the zero-point energy and the free energy in terms of an integral kernel acting on the boundaries; it can be expanded in a convergent series interpreted as a succession of an even number of scatterings of a wave. The quantum and thermal fluctuations of vacuum then appear as a purely geometric property. The Casimir effect thus defined exists only owing to the electromagnetic nature of the field. It does not exist for thin foils with sharp folds, but Casimir forces between solid wedges are finite. We work out various applications: low temperature, high temperature where wrinkling constraints appear, stability of a plane foil, transfer of energy from one side of a curved boundary to the other, forces between distant conductors, special shapes (parallel plates, sphere, cylinder, honeycomb)., 44 pages, 8 figures; Proceedings of the 15 th SIGRAV Conference on General Relativity and Gravitational Physics, Villa Mondragone, Monte Porzio Catone, Roma, Italy, September 9-12, 2002

We consider a model of a D-dimensional tethered manifold interacting by excluded volume in R^d with a single point. Use of intrinsic distance geometry provides a rigorous definition of the analytic continuation of the perturbative expansion for arbitrary D, 0 < D < 2. Its one-loop renormalizability is first established by direct resummation. A renormalization operation R is then described, which ensures renormalizability to all orders. The similar question of the renormalizability of the self-avoiding manifold (SAM) Edwards model is then considered, first at one-loop, then to all orders. We describe a short-distance multi-local operator product expansion, which extends methods of local field theories to a large class of models with non-local singular interactions. It vindicates the direct renormalization method used earlier in part I of these lectures, as well as the corresponding scaling laws., 32 pages, 9 figures, Second Part and extensive update of Lecture Notes originally given in ``Statistical Mechanics of Membranes and Surfaces'', Fifth Jerusalem Winter School for Theoretical Physics (1987), D. R. Nelson, T. Piran,and S. Weinberg eds