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Denis Bernard

Publications and source records attributed to Denis Bernard.

At least 127 records · Page 7Linked to original sources
arXiv2003

CFTs of SLEs: the radial case

We present a relation between conformal field theories (CFT) and radial stochastic Schramm-Loewner evolutions (SLE) similar to that we previously developed for the chordal SLEs. We construct an important local martingale using degenerate representations of the Virasoro algebra. We sketch how to compute derivative exponants and the restriction martingales in this framework. In its CFT formulation, the SLE dual Fokker-Planck operator acts as the two-particle Calogero hamiltonian on boundary primary fields and as the dilatation operator on bulk primary fields localized at the fixed point of the SLE map.

math-ph↗
arXiv2003

Conformal transformations and the SLE partition function martingale

We present an implementation in conformal field theory (CFT) of local finite conformal transformations fixing a point. We give explicit constructions when the fixed point is either the origin or the point at infinity. Both cases involve the exponentiation of a Borel subalgebra of the Virasoro algebra. We use this to build coherent state representations and to derive a close analog of Wick's theorem for the Virasoro algebra. This allows to compute the conformal partition function in non trivial geometries obtained by removal of hulls from the upper half plane. This is then applied to stochastic Loewner evolutions (SLE). We give a rigorous derivation of the equations, obtained previously by the authors, that connect the stochastic Loewner equation to the representation theory of the Virasoro algebra. We give a new proof that this construction enumerates all polynomial SLE martingales. When one of the hulls removed from the upper half plane is the SLE hull, we show that the partition function is a famous local martingale known to probabilists, thereby unravelling its CFT origin.

math-ph↗
arXiv2003

Conformal Field Theories of Stochastic Loewner Evolutions

Stochastic Loewner evolutions (SLE) are random growth processes of sets, called hulls, embedded in the two dimensional upper half plane. We elaborate and develop a relation between SLE evolutions and conformal field theories (CFT) which is based on a group theoretical formulation of SLE processes and on the identification of the proper hull boundary states. This allows us to define an infinite set of SLE zero modes, or martingales, whose existence is a consequence of the existence of a null vector in the appropriate Virasoro modules. This identification leads, for instance, to linear systems for generalized crossing probabilities whose coefficients are multipoint CFT correlation functions. It provides a direct link between conformal correlation functions and probabilities of stopping time events in SLE evolutions. We point out a relation between SLE processes and two dimensional gravity and conjecture a reconstruction procedure of conformal field theories from SLE data.

hep-th↗
arXiv2003

SLE martingales and the Virasoro algebra

We present an explicit relation between representations of the Virasoro algebra and polynomial martingales in stochastic Loewner evolutions (SLE). We show that the Virasoro algebra is the spectrum generating algebra of SLE martingales. This is based on a new representation of the Virasoro algebra, inspired by the Borel-Weil construction, acting on functions depending on coordinates parametrizing conformal maps.

hep-th↗
arXiv2003

A simple asymmetric evolving random network

We introduce a new oriented evolving graph model inspired by biological networks. A node is added at each time step and is connected to the rest of the graph by random oriented edges emerging from older nodes. This leads to a statistical asymmetry between incoming and outgoing edges. We show that the model exhibits a percolation transition and discuss its universality. Below the threshold, the distribution of component sizes decreases algebraically with a continuously varying exponent depending on the average connectivity. We prove that the transition is of infinite order by deriving the exact asymptotic formula for the size of the giant component close to the threshold. We also present a thorough analysis of aging properties. We compute local-in-time profiles for the components of finite size and for the giant component, showing in particular that the giant component is always dense among the oldest nodes but invades only an exponentially small fraction of the young nodes close to the threshold.

cond-mat.dis-nn↗
arXiv2001

Renormalization group for network models of Quantum Hall transitions

We analyze in detail the renormalization group flows which follow from the recently proposed all orders beta functions for the Chalker-Coddington network model. The flows in the physical regime reach a true singularity after a finite scale transformation. Other flows are regular and we identify the asymptotic directions. One direction is in the same universality class as the disordered XY model. The all orders beta function is computed for the network model of the spin Quantum Hall transition and the flows are shown to have similar properties. It is argued that fixed points of general current-current interactions in 2d should correspond to solutions of the Virasoro master equation. Based on this we identify two coset conformal field theories osp(2N|2N)_1 /u(1)_0 and osp(4N|4N)_1/su(2)_0 as possible fixed points and study the resulting multifractal properties. We also obtain a scaling relation between the typical amplitude exponent alpha_0 and the typical point contact conductance exponent X_t which is expected to hold when the density of states is constant.

cond-mat.mes-hall↗
arXiv2001

Strong-weak coupling duality in anisotropic current interactions

The recently proposed all orders beta function is further investigated. By using a strong-weak coupling duality of the beta function, and some added topology of the space of couplings we are able to extend the flows to arbitrarily large or small scales. Using a non-trivial RG invariant we are able to identify sine-Gordon, sinh-Gordon and Kosterlitz-Thouless phases. We also find an additional phase with cyclic or roaming RG trajectories.

hep-th↗
arXiv2000

Turbulence for (and by) amateurs

Series of lectures on statistical turbulence written for amateurs but not experts. Elementary aspects and problems of turbulence in two and three dimensional Navier-Stokes equation are introduced. A few properties of scalar turbulence and transport phenomena in turbulent flows are described. Kraichnan's model of passive advection is discussed a bit more precisely. {Part 1: Approaching turbulent flows.} Navier-Stokes equation. Cascades and Kolmogorov theory. Modeling statistical turbulence. Correlation functions and scaling. {Part 2: Deeper in turbulent flows.} Turbulence in two dimensions. Dissipation and dissipative anomalies. Fokker-Planck equations. Multifractal models. {Part 3: Scalar turbulence.} Transport and Lagrangian trajectories. Kraichnan's passive scalar model. Anomalous scalings and universality. {Part 4: Lagrangian trajectories.} Richardson's law. Lagrangian flows in Kraichnan's model. Slow modes. Breakdown of Lagrangian flows. Batchelor limit. Generalized Lagrangian flows and trajectory bundles.

cond-mat↗
arXiv2000

Influence of Friction on the Direct Cascade of the 2d Forced Turbulence

We discuss two possible scenario for the direct cascade in two dimensional turbulent systems in presence of friction which differ by the presence or not of enstrophy dissipation in the inviscid limit.They are distinguished by the existence or not of a constant enstrophy transfer and by the presence of leading anomalous scaling in the velocity three point functions. We also point out that the velocity statistics become gaussian in the approximation consisting in neglecting odd order correlations in front even order ones.

chao-dyn↗
arXiv2000

Vertex Operator Solutions of 2d Dimensionally Reduced Gravity

We apply algebraic and vertex operator techniques to solve two dimensional reduced vacuum Einstein's equations. This leads to explicit expressions for the coefficients of metrics solutions of the vacuum equations as ratios of determinants. No quadratures are left. These formulas rely on the identification of dual pairs of vertex operators corresponding to dual metrics related by the Kramer-Neugebauer symmetry.

solv-int↗
arXiv1999

Sailing the Deep Blue Sea of Decaying Burgers Turbulence

We study Lagrangian trajectories and scalar transport statistics in decaying Burgers turbulence. We choose velocity fields, solutions of the inviscid Burgers equation, whose probability distributions are specified by Kida's statistics. They are time-correlated, not time-reversal invariant and not Gaussian. We discuss in some details the effect of shocks on trajectories and transport equations. We derive the inviscid limit of these equations using a formalism of operators localized on shocks. We compute the probability distribution functions of the trajectories although they do not define Markov processes. As physically expected, these trajectories are statistically well-defined but collapse with probability one at infinite time. We point out that the advected scalars enjoy inverse energy cascades. We also make a few comments on the connection between our computations and persistence problems.

chao-dyn↗
arXiv1999

On The Three Point Velocity Correlation Functions in 2d Forced Turbulence

We present a simple exact formula for the three point velocity correlation functions in two dimensional turbulence which is valid on all scales and which interpolates between the direct and inverse cascade regimes. As expected, these correlation functions are universal in these extreme regimes. We also discuss the effect of anisotropy and friction.

chao-dyn↗
arXiv1998

Scaling and Exotic Regimes in Decaying Burgers Turbulence

We analyze the stochastic scaling laws arising in the invicid limit of the decaying solutions of the Burgers equation. The linear scaling of the velocity structure functions is shown to reflect the domination by shocks of the long-time asymptotics. We exhibit new self-similar statistics of solutions describing phases with diluted shocks. Some speculations are included on the nature of systems whose large time behavior is described by the new statistics.

chao-dyn↗
arXiv1997

Slow modes in passive advection

The anomalous scaling in the Kraichnan model of advection of the passive scalar by a random velocity field with non-smooth spatial behavior is traced down to the presence of slow resonance-type collective modes of the stochastic evolution of fluid trajectories. We show that the slow modes are organized into infinite multiplets of descendants of the primary conserved modes. Their presence is linked to the non-deterministic behavior of the Lagrangian trajectories at high Reynolds numbers caused by the sensitive dependence on initial conditions within the viscous range where the velocity fields are more regular. Revisiting the Kraichnan model with smooth velocities we describe the explicit solution for the stationary state of the scalar. The properties of the probability distribution function of the smeared scalar in this state are related to a quantum mechanical problem involving the Calogero-Sutherland Hamiltonian with a potential.

cond-mat↗
arXiv1996

Anomalous Scaling in the N-Point Functions of Passive Scalar

A recent analysis of the 4-point correlation function of the passive scalar advected by a time-decorrelated random flow is extended to the N-point case. It is shown that all stationary-state inertial-range correlations are dominated by homogeneous zero modes of singular operators describing their evolution. We compute analytically the zero modes governing the N-point structure functions and the anomalous dimensions corresponding to them to the linear order in the scaling exponent of the 2-point function of the advecting velocity field. The implications of these calculations for the dissipation correlations are discussed.

chao-dyn↗
arXiv1995

(Perturbed) Conformal Field Theory Applied To 2D Disordered Systems: An Introduction

We describe applications of (perturbed) conformal field theories to two-dimensional disordered systems. We present various methods of study~: (i) {\it A direct method} in which we compute the explicit disorder dependence of the correlation functions for any sample of the disorder. This method seems to be specific to two dimensions. The examples we use are disordered versions of the Abelian and non-Abelian WZW models. We show that the disordered WZW model over the Lie group $\CG$ at level $k$ is equivalent at large impurity density to the product of the WZW model over the coset space $\CG^C/\CG$ at level $(-2h^v)$ times an arbitrary number of copies of the original WZW model. (ii) {\it The supersymmetric method} is introduced using the random bond Ising model and the random Dirac theory as examples. In particular, we show that the relevent algebra is the affine $OSp(2N|2N)$ Lie superalgebra, an algebra with zero superdimension. (iii) {\it The replica method} is introduced using the random phase sine-Gordon model as example. We describe particularities of its renormalization group flow. (iv) {\it A variationnal approach} is also presented using the random phase sine-Gordon model as example. Lectures presented at the '95 Cargese Summer School on "Low dimensional application of quantum field theory".

hep-th↗