Bertrand Duplantier

Bertrand Duplantier
Atomic Energy and Alternative Energies Commission | CEA · Institute for Theoretical Physics

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171
Publications
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6,748
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Publications

Publications (171)
Preprint
Full-text available
We present new results for the complex generalized integral means spectrum for two kinds of whole-plane Loewner evolutions driven by L\'evy processes: - L\'evy 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 integra...
Book
This volume provides a detailed description of some of the most active areas in astrophysics from the largest scales probed by the Planck satellite to massive black holes that lie at the heart of galaxies and up to the much awaited but stunning discovery of thousands of exoplanets. It contains the following chapters: • Jean-Philippe UZAN, The Big-...
Article
Full-text available
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...
Preprint
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 \emph{bridge} configuration. We further define multi-bridge networks, where several vertices are in loca...
Preprint
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 $\gamma_b$ to that of terminally-attached self-avoiding arches, $\gamma_{1,1},$ and the {correlation} length exponent $\nu.$ We find $\gamma_b = \gamma_{1,1}+\nu...
Article
Full-text available
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 generali...
Article
Full-text available
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 be...
Chapter
We survey the properties of the log-correlated Gaussian field (LGF), which is a centered Gaussian random distribution (generalized function) h on ℝd , defined up to a global additive constant.
Book
This fifteenth volume of the Poincare Seminar Series, Dirac Matter, describes the surprising resurgence, as a low-energy effective theory of conducting electrons in many condensed matter systems, including graphene and topological insulators, of the famous equation originally invented by P.A.M. Dirac for relativistic quantum mechanics. In five high...
Article
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)...
Book
This fourteenth volume in the Poincaré Seminar Series is devoted to Niels Bohr, his foundational contributions to understanding atomic structure and quantum theory and their continuing importance today. This book contains the following chapters: - Tomas Bohr, Keeping Things Open; - Olivier Darrigol, Bohr's Trilogy of 1913; -John Heilbron, The Mind...
Book
This thirteenth volume of the Poincaré Seminar Series, Henri Poincaré, 1912-2012, is published on the occasion of the centennial of the death of Henri Poincaré in 1912. It presents a scholarly approach to Poincaré’s genius and creativity in mathematical physics and mathematics. Its five articles are also highly pedagogical, as befits their origin i...
Article
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 emb...
Article
Full-text available
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, \phi_1), (h, \phi_2) \bigr] = \int_{\mathbb R^d \times \mathbb R^d} -\lo...
Article
Full-text available
Gaussian Multiplicative Chaos is a way to produce a measure on R[superscript d] (or subdomain of R[superscript d]) of the form e[superscript γX(x)]dx, where X is a log-correlated Gaussian field and γ∈[ 0, √2d) is a fixed constant. A renormalization procedure is needed to make this precise, since X oscillates between −∞ and ∞ and is not a function i...
Article
Full-text available
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 proces...
Article
Full-text available
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 co...
Article
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...
Article
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 m...
Article
Full-text available
Loewner introduced his famous differential equation in 1923 in order to solve Bieberbach conjecture for n=3. His method has been revived in 1999 by Ode Schramm who introduced Stochastic Loewner processes which happened to open many doors in statistical mechanics. The aim of this paper is to revisit Bieberbach conjecture in the framework of SLE and...
Book
This tenth volume in the Poincaré Seminar Series describes recent developments at one of the most challenging frontiers in statistical physics - the deeply related fields of glassy dynamics, especially near the glass transition, and of the statics and dynamics of granular systems. These fields are marked by a vigorous interchange between experiment...
Article
This Proceeding describes joint work [1, 2] with SCOTT SHEFFIELD, and presents a (mathematically rigorous) probabilistic and geometrical proof of the Knizhnik-Polyakov-Zamolodchikov (KPZ) relation between scaling exponents in a Euclidean planar domain D and in Liouville quantum gravity. The Liouville quantum gravity measure on D is the weak limit a...
Article
We present a (mathematically rigorous) probabilistic and geometrical proof of the Knizhnik-Polyakov-Zamolodchikov relation between scaling exponents in a Euclidean planar domain D and in Liouville quantum gravity. It uses the properly regularized quantum area measure dmicro_{gamma}=epsilon;{gamma;{2}/2}e;{gammah_{epsilon}(z)}dz, where dz is the Leb...
Article
Full-text available
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 permutation \sigma of {1,2,...,n} such that p_i=-1 implies \sigma(i)<\sigma(i+1), while p_i=1 implies \sigma(i)>\sigma(i+1). We establish a bijectio...
Article
Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π)−1∫D ∇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 $$\varepsilon^{\gamma^2/2} e^{\gamma h_\varepsilon(z)}dz,$$ where dz is Lebesgue measure on D and h ε (z) denotes t...
Article
Full-text available
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 ext...
Chapter
After recalling the coceptual foundations and the vasic sturcture of general relativity, we review some of its main modern developments (apart from cosmology): ( i ) the post-Newtonian limit and weak-field tests in the solar system, (ii) strong gravitationl feelds and black holes, (iii) strong-field and radiative tests in binary pulsar observations...
Chapter
Full-text available
The Gravity Probe B Relativity Mission was successfully launched on April 20, 2004 from Vandenberg Air Force Base in California, a culmination of 40 years of collaborative development at Stanford University and NASA. The goal of the GP-B experiment is to perform precision tests of two independent predictions of general relativity, the geodetic effe...
Chapter
Gravitatioanl Wave Astronomy progressively becomes this new window on the universe that we expected since tens of years. The technology has now reached a point where large instruments meet a level of sensitivity relevant for astrophysics. Depending on the sector of physics to be addressed, quired. Ground based antennas are already built in Europe,...
Chapter
Full-text available
Despite the impotant experimental success of General Relativity, there are several theoretical reasons indicating that gravitational phenomena may change radically from the predictions of Einstein’s theory at very short distances. A main motivation comes from studies of unifying all fundamental forces in the framework of a consistent quantum theory...
Chapter
A new eaa in fundamental physics began with the discovery of pulsars 1967, the discovery of the first binary pulsar in 1974 and the first millisecond pulsar in 1982. Ever since, pulsars have been used as precise cosmic clocks, taking us beyond the weak-field regime of the solar-system in the study of theories of gravity. Their contribution is cruci...
Article
We present here the results of a direct calculation of the properties of polymer chains near the θ-tricritical point, i.e. in a poor solvent, a problem which has been the subject of contradictory studies. We have devised for this (to all orders) a direct renormalization method. We find, for space dimensions d < 3 and d = 3, new results, and confirm...
Article
We consider a general polymer network in a Θ-solvent. The topology of is arbitrary but fixed. Using the Edwards model with three-body interactions, we calculate the exact asymptotic partition function of in three dimensions at Θ: Θ() ~ S−2/3(lnS)−∑L≥1nL(L−1)(L−2)/132, where S is the common size of the chains, the topological number of loops of and...
Article
A compact and convergent integral representation is established for any dimensionally renormalized d.r. polymer partition function. New rules for calculating the diagrams of the Fixman expansion yield directly their finite d.r. part. An operation R' acts on the integrand, the same as devised by Bergère and David for field-theoretic Feynman amplitud...
Article
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...
Book
The Poincaré Seminar is held twice a year at the Institute Henri Poincaré in Paris. The goal of this seminar is to provide up-to-date information about general topics of great interest in physics. Both the theoretical and experimental results are covered, with some historical background. Particular care is devoted to the pedagogical nature of the p...
Article
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...
Book
The Genesis of the Theory of Relativity.- Special Relativity: A Centenary Perspective.- From Euclid's Geometry to Minkowski's Spacetime.- The de Sitter and anti-de Sitter Sightseeing Tour.- Experiments with Single Photons.- Einstein 1905-1955: His Approach to Physics.- On Boltzmann's Principle and Some Immediate Consequences Thereof.- Brownian Moti...
Article
Full-text available
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...
Article
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 reno...
Article
The exact joint multifractal distribution for the scaling and spiraling of electrostatic potential lines near any conformally invariant scaling curve is derived in two dimensions. Its spectrum f(α,λ) gives the Hausdorff dimension of the points where the potential scales with distance r as\(H \sim {r^\alpha }\)while the curve spirals logarithmically...
Article
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 with interacting random paths, by exploiting an underlying quantum gravity (QG) structure, which uses KPZ maps relating exponents in the plane to those...
Article
Full-text available
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 f(alpha,lambda) gives the Hausdorff dimension of the points where the potential scales with distance r as H approximately r(alpha) while the curve logarith...
Chapter
Constante de Planck: $ h = h/2\pi \simeq 1,055 \times {10^{-34}}Js $ h = h/2\pi \simeq 1,055 \times {10^{-34}}Js ; vitesse de la lumière: $ c \simeq 3 \times {10^8}m{s^{ - 1}} $ c \simeq 3 \times {10^8}m{s^{ - 1}} constante de Boltzmann: $ {k_B} \simeq 1,381 \times {10^{ - 23}}J{K^{ - 1}} $ {k_B} \simeq 1,381 \times {10^{ - 23}}J{K^{ - 1}} D’après...
Article
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 fractal in two dimensions, corresponding to a conformal field theory of central charge c. It gives the Hausdorff dimension of the set of boundary point...
Article
The multifractal distribution of the potential near any conformally invariant fractal boundary, like a critical Ising, or percolation, or Potts cluster, or like a random or a self-avoiding walk, is solved exactly in two dimensions. The multifractal dimension hatf(theta) of the boundary set with local opening wedge angle theta is hatf(theta)=fracpit...
Article
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 dimensions. The dimension &fcirc;(straight theta) of the boundary set with local wedge angle straight theta is &fcirc;(straight theta) = pi / straight theta-25-c /...
Article
The harmonic measure (or diffusion field) near a critical percolation cluster in two dimensions (2D) is considered. Its moments, summed over the accessible external hull, exhibit a multifractal (MF) spectrum, which I calculate exactly. The generalized dimensions D\(n\) as well as the MF function f\(alpha\) are derived from generalized conformal inv...
Article
We consider L planar random walks (or Brownian motions) of large length, t, starting at neighboring points, and the probability PL (t) ∼ t−ζL that their paths do not intersect. By a 2D quantum gravity method, i.e., a non linear map onto a random Riemann surface, the former conjecture that is established. This also applies to the half-plane where ,...
Article
Full-text available
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 each type. We relate the probabilities rigorously to amplitudes of $O(N=1)$ models whose exponents, believed to be exact, yield $x^{\cal P}_{\ell}=({\ell}...
Article
We present a theoretical study of the interaction of tight DNA crossovers with eukaryotic type II DNA topoisomerases. A quantitative analysis of the role of the enzyme during anaphase first shows that a tight DNA crossover should be an intermediate of the strand-passage reaction. We then focus on the initial steps of the strand-passage reaction in...
Article
Several experimental data support the notion that the recognition of DNA crossovers play an important role in the multiple functions of topoisomerase II. Here, a theoretical analysis of the possible modes of assembly of yeast topoisomerase II with right and left-handed tight DNA crossovers is performed, using the crystal coordinates of the docking...
Article
82B27 Critical phenomena 82B41 Random walks, random surfaces, lattice animals, etc. (See also 60G50, 82C41) 82D60 Polymers
Article
A grand canonical ensemble of continuous chains and rings is introduced for describing polymerisation equilibrium like that of sulphur. Equivalence is shown to O(n) field theory, with n continuous, n to 1.
Article
The author shows that the 'hull percolation' problem considered recently by Roux et al. (1988) is in one-to-one correspondence with standard bond percolation. Their random tiling of the plane by squares with circles is identical with the Baxter-Kelland-Wu polygon decomposition associated with clusters of the Q-state Potts model, which in the Q to 1...
Article
We address the problem of the algebraic area enclosed by a Brownian curve in two dimensions, recently reconsidered by D. C. Khandekar and F. W. Wiegel [ibid. 21, No.10, L 563-L 566 (1988; Zbl 0657.60098)]. We recall the principal results actually first obtained by P. Lévy [C. r. Acad. Sci., Paris 230, 432-434 (1950; Zbl 0034.072)]. Another derivati...
Article
The author’s aim is to derive from conformal invariance the multifractal spectrum of the harmonic measure near a random fractal, such as the frontier of a random walk, i.e., a Brownian motion, a self-avoiding walk, or a percolation cluster. First we consider the related problem of L planar random walks (or Brownian motions) of large time t, startin...
Article
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 dimensions in the plane are all derived from an algebraic structure existing on a random lattice (2D quantum gravity). The multifractal dimensions of t...
Article
We consider L planar random walks (or Brownian motions) of large length t, starting at neighboring points, and the probability PL\(t\)~t-zetaL that their paths do not intersect. By a 2D quantum gravity method, i.e., a nonlinear map to an exact solution on a random surface, I establish our former conjecture that zetaL = 124\(4L2-1\). This also appli...
Article
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 the methods of local field theories to a large class of models with non-local singular interactions. This ensures the existence of scaling laws for crum...
Article
We use a recently developed a priori theory of diffusion-limited aggregation (DLA) to compute multifractal dimensions and their fluctuations, using methods analogous to field theoretical resummations. There are two regimes, depending upon n, the number of particles in the DLA cluster, as well as on the multifractal moment q. In the strongly fluctua...
Article
In the course of anaphase, the chromosomal DNA is submitted to the traction of the spindle. Several physical problems are associated with this action. In particular, the sister chromatids are generally topologically intertwined at the onset of anaphase, and the removal of the intertwinings results from a coupling between the enzymatic action of typ...
Article
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 averaging is needed in order to define multifractal dimensions. In an earlier work [T. C. Halsey and M. Leibig, Phys. Rev. A46, 7793 (1992)], annealed avera...
Article
Sister chromatids are topologically intertwined at the onset of anaphase: their segregation during anaphase is known to require strand-passing activity by type II DNA topoisomerase. We propose that the removal of the intertwinings involves at the same time the traction of the mitotic spindle and the activity of topoisomerases. This implies that the...
Conference Paper
Full-text available
This text provides an overview of my research on DNA in 1995: - Structural transitions in DNA: helix-coil, coil-globule and isotropic to anisotropic (liquid crytalline DNA); coupling between transitions; kinetics of renaturation. - Topological constraints in DNA and chromosomes and DNA topoisomerases: transition from reptation to Rouse relaxation,...
Article
The statistics of a long closed self-avoiding walk (SAW) or polymer ring on a d-dimensional lattice obeys hyperscaling. The combination (where pN is the number of configurations of an oriented and rooted N-step ring, 〈R2〉N a typical average size squared, and μ the SAW effective connectivity constant of the lattice) is equal for to a lattice-depende...
Article
The renormalizability of the self-avoiding manifold (SAM) Edwards model is established. We use a new short distance multilocal operator product expansion (MOPE), which extends methods of local field theories to a large class of models with non-local singular interactions. This validates the direct renormalization method introduced before, as well a...
Article
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 form: $$ Q_{\ell} \sim \mu^{ \ell}_ 0\mu^{ \ell^ \sigma}_ 1\ell^{ \gamma -1}\ , \eqno $$ where $ \mu_ 0(T) $ and $ \mu_ 1(T) $ are respectively bulk and...
Article
A Comment on the Letter by A. T. Boothroyd et al., Phys. Rev. Lett. 69, 426 (1992).
Article
We consider a model of D-dimensional tethered manifold interacting by excluded volume in R^d with a single point. By use of intrinsic distance geometry, we first provide a rigorous definition of the analytic continuation of its perturbative expansion for arbitrary D, 0 < D < 2. We then construct explicitly a renormalization operation, ensuring reno...
Article
We consider a continuous model of D-dimensional elastic (polymerized) manifold fluctuating in d-dimensional euclidean space, interacting with a single impurity via an attractive or repulsive δ-potential (but without self-avoidance interactions). Except for D = 1 (the polymer case), this model cannot be mapped onto a loca; field theory. We show that...
Article
Loop-erased self-avoiding walks (LESAWs) are defined as walks resulting from sequential erasing of the loops of random walks. The critical properties of LESAWs are those of a c = -2 conformal theory in two dimensions, and, geometrically, those of subgraphs of spanning trees. In this paper, we derive the exact asymptotic behavior of the set of proba...
Article
We consider polymer networks of arbitrary but fixed topology. We prove the multiplicative renormalizability of these networks in terms of their star-like vertices by relating it to special field theoretic correlation functions. To avoid additive renormalizations these correlation functions have to be differentiated with respect to certain temperatu...
Article
Full-text available
We show that the boundary conditions imposed on the director fluctuations in nematics by the presence of rigid walls give rise to long-range forces analogous to the Casimir effect in electrodynamics. We discuss different calculational schemes for the derivation of this results. We derive the spatial behavior of this interaction for smectics and col...
Article
We consider [B. Duplantier et al., Phys. Rev. Lett.65, 508 (1990); B. Duplantier, PhysicaA168, 179 (1990)] charged (and possibly conducting) membranes of arbitrary shapes immersed in an ionic solution. The electrostatic double layer contribution to their free energy is exactly expressed within the Debye—Hückel approximation as a convergent multiple...
Article
B. Duplantier gives reply to the comments made by G. A. Baker on his work on thermodynamic properties of colloid surface.(AIP)

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