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1 Observatoire de la Côte d’Azur, Lab. G.D. Cassini, B.P. 4229, F-06304 Nice Cedex 4, France. E-mail: bec@obs-nice.fr 2 Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, UK. E-mail: K.Khanin@ma.hw.ac.uk 3 Isaac Newton Institute for Mathematical Sciences, 20 Clarkson Road, Cambridge CB3 0EH, UK. 4 Landau Institute for Theoretical… (More)

We introduce a new disorder regime for directed polymers in dimension 1 + 1 that sits between the weak and strong disorder regimes. We call it the intermediate disorder regime. It is accessed by scaling the inverse temperature parameter β to zero as the polymer length n tends to infinity. The natural choice of scaling is βn := βn−1/4. We show that the… (More)

We prove convergence of stationary distributions for the randomly forced Burgers and Hamilton-Jacobi equations in a limit when viscosity tends to zero. It turns out that for all values of the viscosity ν there exists a unique (up to an additive constant) solution to the randomly forced Hamilton-Jacobi equation which is extendible for all times. The main… (More)

We prove the renormalization conjecture for circle diffeomorphisms with breaks, i.e., that the renormalizations of any two C2+α-smooth (α ∈ (0, 1)) circle diffeomorphisms with a break point, with the same irrational rotation number and the same size of the break, approach each other exponentially fast in the C2-topology. As was shown in [18], this result… (More)

The main aim of this paper is to present a simple probabilistic model for the early stage of neuron growth: the speciication on an axon out of several initially similar neurites. The model is a Markov process with competition between the growing neu-rites, wherein longer objects have more chances to grow, and parameter determines the intensity of the… (More)

- Konstantin Khanin, Saša Kocić, Elio Mazzeo
- 2011

We prove that for almost all rotation numbers, every two C2+α-smooth circle diffeomorphisms with a break point, with the same irrational rotation number and the same size of the break, are C1-smoothly conjugate to each other, provided that the corresponding renormalizations approach asymptotically each other with an exponential rate.

We consider directed polymers in a random potential given by a deterministic profile with a strong maximum at the origin taken with random sign at each integer time. We study two main objects based on paths in this random potential. First, we use the random potential and averaging over paths to define a parabolic model via a random Feynman–Kac evolution… (More)

- Konstantin Khanin, Sergei Nechaev, Gleb Oshanin, Andrei Sobolevski, Oleg Vasilyev
- Physical review. E, Statistical, nonlinear, and…
- 2010

We consider a (1+1)-dimensional ballistic deposition process with next-nearest-neighbor interactions, which belongs to the Kardar-Parisi-Zhang (KPZ) universality class. The focus of our analysis is on the properties of structures appearing in the bulk of a growing aggregate: a forest of independent clusters separated by "crevices." Competition for growth… (More)

In the present note we announce a proof of a strong form of Arnold diffusion for smooth convex Hamiltonian systems. Let T2 be a 2-dimensional torus and B2 be the unit ball around the origin inR2. Fix ρ > 0. Our main result says that for a ‘generic’ time-periodic perturbation of an integrable system of two degrees of freedom H0(p) + εH1(θ, p, t), θ ∈ T2, p ∈… (More)

For almost all irrational ρ ∈ (0, 1), any two cyclic generalized interval exchange transformations with breaks on the same orbit, with the same rotation number ρ, and the same size of the corresponding breaks, are C1-smoothly conjugate to each other. In particular, for almost all irrational ρ ∈ (0, 1), generalized interval exchange transformations of two… (More)