A PATCHWORK QUILT SEWN FROM BROWNIAN FABRIC: REGULARITY OF POLYMER WEIGHT PROFILES IN BROWNIAN LAST PASSAGE PERCOLATION

In last passage percolation models lying in the Kardar–Parisi–Zhang (KPZ) universality class, the energy of long energy-maximizing paths may be studied as a function of the paths’ pair of endpoint locations. Scaled coordinates may be introduced, so that these maximizing paths, or polymers, now cross unit distances with unit-order fluctuations, and have scaled energy, or weight, of unit order. In this article, we consider Brownian last passage percolation in these scaled coordinates. In the narrow wedge case, when one endpoint of such polymers is fixed, say at $(0,0)\in \mathbb{R}^{2}$ , and the other is varied horizontally, over $(z,1)$ , $z\in \mathbb{R}$ , the polymer weight profile as a function of $z\in \mathbb{R}$ is locally Brownian; indeed, by Hammond [‘Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation’, Preprint (2016), arXiv:1609.02971, Theorem 2.11 and Proposition 2.5], the law of the profile is known to enjoy a very strong comparison to Brownian bridge on a given compact interval, with a Radon–Nikodym derivative in every $L^{p}$ space for $p\in (1,\infty )$ , uniformly in the scaling parameter, provided that an affine adjustment is made to the weight profile before the comparison is made. In this article, we generalize this narrow wedge case and study polymer weight profiles begun from a very general initial condition. We prove that the profiles on a compact interval resemble Brownian bridge in a uniform sense: splitting the compact interval into a random but controlled number of patches, the profile in each patch after affine adjustment has a Radon–Nikodym derivative that lies in every $L^{p}$ space for $p\in (1,3)$ . This result is proved by harnessing an understanding of the uniform coalescence structure in the field of polymers developed in Hammond [‘Exponents governing the rarity of disjoint polymers in Brownian last passage percolation’, Preprint (2017a), arXiv:1709.04110] using techniques from Hammond (2016) and [‘Modulus of continuity of polymer weight profiles in Brownian last passage percolation’, Preprint (2017b), arXiv:1709.04115].

Initialization of Fractional-Order Systems Using the Hankel Operator

This paper demonstrates the use of the Hankel operator to characterize the initial-condition response of a fractional-order system. A general initial-condition response can be determined for any input applied before t = 0. Two techniques for approximating the Hankel operator are discussed. These approximation methods are applied to illustrative examples, demonstrating a general characterization of natural responses.

A GENERAL INITIAL CONDITION OF INFLATIONARY COSMOLOGY ON TRANS-PLANCKIAN PHYSICS

We consider a more general initial condition satisfying the minimal uncertainty relationship. We calculate the power spectrum of a simple model in inflationary cosmology. The results depend on perturbations generated below a fundamental scale, e.g. the Planck scale.

Dynamics of the Electronic States of Organic Charge-Transfer Salts in in-Phase Mode

We study the dimer charge oscillations of organic charge-transfer salts in in-phase mode due to the general initial condition problem. The exact time evolution of the half-filled (two-particle) case has a complex behavior. Under certain initial conditions, the average number of fermions in one-particle state, in the half-filled case, is not necessarily periodic, even though the magnetization remains periodic. We present the expressions of the magnetization and the transition dipole moment of the organic charge-transfer salts in the in-phase mode coupled to a constant external magnetic field, for arbitrary initial conditions. We show that the constant external magnetic field plays no role in this phenomena.