Anomalous 1D fluctuations of a simple 2D random walk in a large deviation regime

Abstract

The following question is the subject of our work: could a two-dimensional random path pushed by some constraints to an improbable "large deviation regime", possess extreme statistics with one-dimensional Kardar-Parisi-Zhang (KPZ) fluctuations? The answer is positive, though non-universal, since the fluctuations depend on the underlying geometry. We consider in details two examples of 2D systems for which imposed external constraints force the underlying stationary stochastic process to stay in an atypical regime with anomalous statistics. The first example deals with the fluctuations of a stretched 2D random walk above a semicircle or a triangle. In the second example we consider a 2D biased random walk along a channel with forbidden voids of circular and triangular shapes. In both cases we are interested in the dependence of a typical span $⟨d(t)⟩\sim t^{\gamma }$ of the trajectory of $t$ steps above the top of the semicircle or the triangle. We show that $\gamma =\frac{1}{3}$, i.e. $⟨d(t)⟩$ shares the KPZ statistics for the semicircle, while $\gamma =0$ for the triangle. We propose heuristic derivations of scaling exponents $\gamma$ for different geometries, justify them by explicit analytic computations and compare with numeric simulations. For practical purposes, our results demonstrate that the geometry of voids in a channel might have a crucial impact on the width of the boundary layer and, thus, on the heat transfer in the channel.

17 pages, 8 figures, some parts of the paper are rewritten

References in corpus (14)

• Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
• Discrete polynuclear growth and determinantal processes
• Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers
• Exact distribution of the maximal height of p vicious walkers
• Scale Invariance of the PNG Droplet and the Airy Process
• Exact Asymptotic Results for a Model of Sequence Alignment
• Constrained Brownian motion: Fluctuations away from circular and parabolic barriers
• Limiting distributions for a polynuclear growth model with external sources
• Geometrical optics of constrained Brownian excursion: from the KPZ scaling to dynamical phase transitions
• An invariance principle to Ferrari-Spohn diffusions
• Statistics of the maximal distance and momentum in a trapped Fermi gas at low temperature
• Replica Bethe ansatz derivation of the GOE Tracy-Widom distribution in one-dimensional directed polymers with free boundary conditions
• Bethe Ansatz in the Bernoulli Matching Model of Random Sequence Alignment
• Winding angle distribution for planar random walk, polymer ring entangled with an obstacle, and all that: Spitzer-Edwards-Prager-Frisch model revisited

Cited by in corpus (13)

• The Airy distribution: experiment, large deviations and additional statistics
• Geometrical optics of constrained Brownian motion: three short stories
• Geometrical optics of constrained Brownian excursion: from the KPZ scaling to dynamical phase transitions
• Non-crossing Brownian paths and Dyson Brownian motion under a moving boundary
• Quantum information scrambling in non-Markovian open quantum systems
• Brownian flights over a circle
• Field-driven tracer diffusion through curved bottlenecks: Fine structure of first passage events
• Geometrical optics of large deviations of Brownian motion in inhomogeneous media
• Fixman problem revisited: When fluctuations of inflated ideal polymer loop are non-Gaussian?
• Stretching of a fractal polymer around a disc reveals KPZ-like statistics
• Constrained non-crossing Brownian motions, fermions and the Ferrari-Spohn distribution
• Lifshitz tails at spectral edge and holography with a finite cutoff
• Non-algebraic first return probability of a stretched random walk near a convex boundary and its effect on adsorption