Pierre Le Doussal

Self-consistent theory of polymerized membranes

We study D-dimensional polymerized membranes embedded in d dimensions using a self-consistent screening approximation. It is exact for large d to order 1/d, for any d to order \ensuremath{\epsilon}=4-D, and for d=D. For flat physical membranes (D=2, d=3) it predicts a roughness exponent \ensuremath{\zeta}=0.590. For phantom membranes at the crumpling transition the size exponent is \ensuremath{\nu}=0.732. It yields identical lower critical dimension for the flat phase and crumpling transition

Exact solution for the Kardar-Parisi-Zhang equation with flat initial conditions.

{\mathit{D}}_{\mathrm{lc}}

The KPZ equation with flat initial condition and the directed polymer with one free end

(d)=2d/(d+1) (

From equilibrium spin models to probabilistic cellular automata

{\mathit{D}}_{\mathrm{lc}}

Numerical study of aD-dimensional periodic Lorentz gas with universal properties

= \ensuremath{\surd}2 for codimension 1). For physical membranes with random quenched curvature \ensuremath{\zeta}=0.775 in the new T=0 flat phase in good agremeent with simulations.

Driven particle in a random landscape: disorder correlator, avalanche distribution, and extreme value statistics of records.

We provide the first exact calculation of the height distribution at arbitrary time t of the continuum Kardar-Parisi-Zhang (KPZ) growth equation in one dimension with flat initial conditions. We use the mapping onto a directed polymer with one end fixed, one free, and the Bethe ansatz for the replicated attractive boson model. We obtain the generating function of the moments of the directed polymer partition sum as a Fredholm Pfaffian. Our formula, valid for all times, exhibits convergence of the free energy (i.e., KPZ height) distribution to the Gaussian orthogonal ensemble Tracy-Widom distribution at large time.

Extreme value statistics from the real space renormalization group: Brownian motion, Bessel processes and continuous time random walks

We study the directed polymer (DP) of length t in a random potential in dimension 1 + 1 in the continuum limit, with one end fixed and one end free. This maps onto the Kardar?Parisi?Zhang growth equation in time t, with flat initial conditions. We use the Bethe ansatz solution for the replicated problem, which is an attractive bosonic model. The problem is more difficult than the previous solution of the fixed endpoint problem as it requires regularization of the spatial integrals over the Bethe eigenfunctions. We use either a large fixed system length or small finite slope KPZ initial conditions (wedge). The latter allows one to take properly into account non-trivial contributions, which appear as deformed strings in the former. By considering a half-space model in a proper limit we obtain an expression for the generating function of all positive integer moments of the directed polymer partition function. We obtain the generating function of the moments of the DP partition sum as a Fredholm Pfaffian. At large time, this Fredholm Pfaffian, valid for all time t, exhibits convergence of the free energy (i.e.?KPZ height) distribution to the GOE Tracy?Widom distribution.

Functional renormalization group at large N for disordered systems.

The general equivalence betweenD-dimensional probabilistic cellular automata (PCA) and (D+1)-dimensional equilibrium spin models satisfying a “disorder condition” is first described in a pedagogical way and then used to analyze the phase diagrams, the critical behavior, and the universality classes of some automata. Diagrammatic representations of time-dependent correlation functions of PCA are introduced. Two important classes of PCA are singled out for which these correlation functions simplify: (1) “Quasi-Hamiltonian” automata, which have a current-carrying steady state, and for which some correlation functions are those of aD-dimensional static model. PCA satisfying the detailed balance condition appear as a particular case of these rules for which the current vanishes. (2) “Linear” (and more generally “affine”) PCA for which the diagrammatics reduces to a random walk problem closely related to (D+1)-dimensional directed SAWs: both problems display a critical behavior with mean-field exponents in any dimension. The correlation length and effective velocity of propagation of excitations can be calculated for affine PCA, as is shown on an explicitD=1 example. We conclude with some remarks on nonlinear PCA, for which the diagrammatics is related to reaction-diffusion processes, and which belong in some cases to the universality class of Reggeon field theory.

Directed polymer near a hard wall and KPZ equation in the half-space

We give the results of a numerical study of the motion of a point particle in ad-dimensional array of spherical scatterers (Sinais billiard without horizon). We find a simple universal law for the Lyapounov exponent (as a function ofd) and a stretched exponential decay for the velocity autocorrelation as a function of the number of collisions. The diffusion seems to be anomalous in this problem. Ergodicity is used to predict the shape of the probability distribution of long free paths. Physical interpretations or clues are proposed.

Avalanche-size distribution at the depinning transition: A numerical test of the theory

We review how the renormalized force correlator Delta(micro) , the function computed in the functional renormalization-group (RG) field theory, can be measured directly in numerics and experiments on the dynamics of elastic manifolds in the presence of pinning disorder. We show how this function can be computed analytically for a particle dragged through a one-dimensional random-force landscape. The limit of small velocity allows one to access the critical behavior at the depinning transition. For uncorrelated forces one finds three universality classes, corresponding to the three extreme value statistics, Gumbel, Weibull, and Fréchet. For each class we obtain analytically the universal function Delta(micro) , the corrections to the critical force, and the joint probability distribution of avalanche sizes s and waiting times w . We find P(s)=P(w) for all three cases. All results are checked numerically. For a Brownian force landscape, known as the Alessandro, Beatrice, Bertotti, and Montorsi (ABBM) model, avalanche distributions and Delta(micro) can be computed for any velocity. For two-dimensional disorder, we perform large-scale numerical simulations to calculate the renormalized force correlator tensor Delta_{ij}(micro[over ]) , and to extract the anisotropic scaling exponents zeta_{x}>zeta_{y} . We also show how the Middleton theorem is violated. Our results are relevant for the record statistics of random sequences with linear trends, as encountered, e.g., in some models of global warming. We give the joint distribution of the time s between two successive records and their difference in value w .