A Bijection between well-labelled positive paths and matchings (2009)
Bernardi, Olivier, Duplantier, Bertrand, Nadeau, Philippe
A well-labelled positive path of size n is a pair (p,\sigma) made of a word p=p_1p_2...p_{n-1} on the alphabet {-1, 0,+1} such that the sum of the letters of any prefix is non-negative, together with...
Gravitation and experiment (2009)
Damour, Thibault, Damour, Thibault, Duplantier, Bertrand, Rivasseau, Vincent
Duality and KPZ in Liouville Quantum Gravity (2009)
Duplantier, Bertrand, Sheffield, Scott
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...
A Bijection between well-labelled positive paths and matchings (2009)
Bernardi, Olivier, Duplantier, Bertrand, Nadeau, Philippe
A well-labelled positive path of size n is a pair (p,\sigma) made of a word p=p_1p_2...p_{n-1} on the alphabet {-1, 0,+1} such that the sum of the letters of any prefix is non-negative, together with...
A Bijection between well-labelled positive paths and matchings (2009)
Bernardi, Olivier, Duplantier, Bertrand, Nadeau, Philippe
A well-labelled positive path of size n is a pair (p,\sigma) made of a word p=p_1p_2...p_{n-1} on the alphabet {-1, 0,+1} such that the sum of the letters of any prefix is non-negative, together with...
Liouville Quantum Gravity and KPZ (2008)
Duplantier, Bertrand, Sheffield, Scott
Consider a bounded planar domain D, an instance h of the Gaussian free field on D (with Dirichlet energy normalized by 1/(2\pi)), and a constant 0 < gamma < 2. The Liouville quantum gravity measure...
Harmonic measure and winding of random conformal paths: A Coulomb gas perspective (2008)
Duplantier, Bertrand, Binder, Ilia
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...
Brownian Motion, "Diverse and Undulating" (2007)
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...
Conformal Random Geometry (2006)
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...
6th Poincaré Seminar on the Quantum Hall Effect (2006)
Douçot, Benoit, Duplantier, Bertrand, Pasquier, Vincent, Rivasseau, Vincent
Geometry of the Casimir Effect (2005)
Balian, Roger, Duplantier, Bertrand
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,...
Geometry of the Casimir Effect (2004)
Balian, Roger, Duplantier, Bertrand
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,...
Statistical Mechanics of Self-Avoiding Manifolds (Part II) (2004)
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...
Geometry of the Casimir Effect (2004)
Balian, Roger, Duplantier, Bertrand
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,...
Geometry of the Casimir Effect (2004)
Balian, Roger, Duplantier, Bertrand
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,...
Conformal Fractal Geometry and Boundary Quantum Gravity (2003)
This article gives a comprehensive description of the fractal geometry of conformally-invariant (CI) scaling curves, in the plane or half-plane. It focuses on deriving critical exponents associated...
Harmonic Measure and Winding of Conformally Invariant Curves (2002)
Duplantier, Bertrand, Binder, Ilia A.
The exact joint multifractal distribution for the scaling and winding of the electrostatic potential lines near any conformally invariant scaling curve is derived in two dimensions. Its spectrum...
Higher Conformal Multifractality (2002)
We derive, from conformal invariance and quantum gravity, the multifractal spectrum f(alpha,c) of the harmonic measure (or electrostatic potential, or diffusion field) near any conformally invariant...
Conformally Invariant Fractals and Potential Theory (1999)
The multifractal (MF) distribution of the electrostatic potential near any conformally invariant fractal boundary, like a critical O(N) loop or a $Q$ -state Potts cluster, is solved in two...
Path Crossing Exponents and the External Perimeter in 2D Percolation (1999)
Aizenman, Michael, Duplantier, Bertrand, Aharony, Amnon
2D Percolation path exponents $x^{\cal P}_{\ell}$ describe probabilities for traversals of annuli by $\ell$ non-overlapping paths, each on either occupied or vacant clusters, with at least one of...
Exact Multifractal Exponents for Two-Dimensional Percolation (1999)
The harmonic measure (or diffusion field or electrostatic potential) near a percolation cluster in two dimensions is considered. Its moments, summed over the accessible external hull, exhibit a...
Two-Dimensional Copolymers and Exact Conformal Multifractality (1998)
We consider in two dimensions the most general star-shaped copolymer, mixing random (RW) or self-avoiding walks (SAW) with specific interactions thereof. Its exact bulk or boundary conformal scaling...
Renormalization Theory for the Self-Avoiding Polymerized Membranes (1997)
David, Francois, Duplantier, Bertrand, Guitter, Emmanuel
We prove the renormalizability of the generalized Edwards model for self-avoiding polymerized membranes. This is done by use of a short distance multilocal operator product expansion, which extends...
Multifractal Dimensions for Branched Growth (1996)
Halsey, Thomas C., Honda, Katsuya, Duplantier, Bertrand
A recently proposed theory for diffusion-limited aggregation (DLA), which models this system as a random branched growth process, is reviewed. Like DLA, this process is stochastic, and ensemble...
Hyperscaling for polymer rings (1994)
The statistics of a long closed self-avoiding walk (SAW) or polymer ring on a $ d $-dimensional lattice obeys hyperscaling. The combination $ p_N \left\langle R^2 \right\rangle^{ d/2}_N\mu^{ -N}, $...
Exact scaling form for the collapsed 2D polymer phase (1993)
It has been recently argued that interacting self-avoiding walks (ISAW) of length $ \ell , $ in their low temperature phase (i.e. below the $ \Theta $-point) should have a partition function of the...