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.Comment: 33 pages, 3 figures, to appear in J. Stat. Phy

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.Comment: 18 pages, one figure in eps format, Latex, published versio

Asymmetric evolving random networks

We generalize the poissonian evolving random graph model of Bauer and Bernard to deal with arbitrary degree distributions. The motivation comes from biological networks, which are well-known to exhibit non poissonian degree distribution. A node is added at each time step and is connected to the rest of the graph by oriented edges emerging from older nodes. This leads to a statistical asymmetry between incoming and outgoing edges. The law for the number of new edges at each time step is fixed but arbitrary. Thermodynamical behavior is expected when this law has a large time limit. Although (by construction) the incoming degree distributions depend on this law, this is not the case for most qualitative features concerning the size distribution of connected components, as long as the law has a finite variance. As the variance grows above 1/4, the average being <1/2, a giant component emerges, which connects a finite fraction of the vertices. Below this threshold, the distribution of component sizes decreases algebraically with a continuously varying exponent. The transition is of infinite order, in sharp contrast with the case of static graphs. The local-in-time profiles for the components of finite size allow to give a refined description of the system.Comment: 30 pages, 3 figure

Loewner Chains

These lecture notes on 2D growth processes are divided in two parts. The first part is a non-technical introduction to stochastic Loewner evolutions (SLEs). Their relationship with 2D critical interfaces is illustrated using numerical simulations. Schramm's argument mapping conformally invariant interfaces to SLEs is explained. The second part is a more detailed introduction to the mathematically challenging problems of 2D growth processes such as Laplacian growth, diffusion limited aggregation (DLA), etc. Their description in terms of dynamical conformal maps, with discrete or continuous time evolution, is recalled. We end with a conjecture based on possible dendritic anomalies which, if true, would imply that the Hele-Shaw problem and DLA are in different universality classes.Comment: 46 pages, 21 figure

On Root Multiplicities of Some Hyperbolic Kac-Moody Algebras

Using the coset construction, we compute the root multiplicities at level three for some hyperbolic Kac-Moody algebras including the basic hyperbolic extension of $A_1^{(1)}$ and $E_{10}$.Comment: 10 pages, LaTe

Local Detailed Balance : A Microscopic Derivation

Thermal contact is the archetype of non-equilibrium processes driven by constant non-equilibrium constraints when the latter are enforced by reservoirs exchanging conserved microscopic quantities. At a mesoscopic scale only the energies of the macroscopic bodies are accessible together with the configurations of the contact system. We consider a class of models where the contact system, as well as macroscopic bodies, have a finite number of possible configurations. The global system with only discrete degrees of freedom has no microscopic Hamiltonian dynamics, but it is shown that, if the microscopic dynamics is assumed to be deterministic and ergodic and to conserve energy according to some specific pattern, and if the mesoscopic evolution of the global system is approximated by a Markov process as closely as possible, then the mesoscopic transition rates obey three constraints. In the limit where macroscopic bodies can be considered as reservoirs at thermodynamic equilibrium (but with different intensive parameters) the mesoscopic transition rates turn into transition rates for the contact system and the third constraint becomes local detailed balance ; the latter is generically expressed in terms of the microscopic exchange entropy variation, namely the opposite of the variation of the thermodynamic entropy of the reservoir involved in a given microscopic jump of the contact system configuration. For a finite-time evolution after contact has been switched on we derive a fluctuation relation for the joint probability of the heat amounts received from the various reservoirs. The generalization to systems exchanging energy, volume and matter with several reservoirs, with a possible conservative external force acting on the contact system, is given explicitly.Comment: 26 pages. arXiv admin note: substantial text overlap with arXiv:1302.453