arXiv subjects

Christian Noack

Publications and source records attributed to Christian Noack.

6 recordsLinked to original sources

arXiv2021

Characterizations of the generalized inverse Gaussian, asymmetric Laplace, and shifted (truncated) exponential laws via independence properties

We prove three new characterizations of the generalized inverse Gaussian (GIG), asymmetric Laplace (AL), shifted exponential (sExp) and shifted truncated exponential (stExp) distributions in terms of non-trivial independence preserving transformations, which were conjectured by Croydon and Sasada in \cite{CS1}. We do this under the assumptions of absolute continuity and mild regularity conditions on the densities. Croydon and Sasada \cite{CS2} use these independence preserving transformations to analyze statistical mechanical models which display KPZ behavior. Our characterizations show the integrability of these models only holds for these four specific distributions in the absolutely continuous setting.

math.PR↗

arXiv2020

KPZ-type fluctuation exponents for interacting diffusions in equilibrium

We consider systems of $N$ diffusions in equilibrium interacting through a potential $V$. We study a "height function" which for the special choice $V(x) = \e^{-x}$, coincides with the partition function of a stationary semidiscrete polymer, also known as the (stationary) O'Connell-Yor polymer. For a general class of smooth convex potentials (generalizing the O'Connell-Yor case), we obtain the order of fluctuations of the height function by proving matching upper and lower bounds for the variance of order $N^{2/3}$, the expected scaling for models lying in the KPZ universality class. The models we study are not expected to be integrable and our methods are analytic and non-perturbative, making no use of explicit formulas or any results for the O'Connell-Yor polymer.

math.PR↗

arXiv2020

Concentration for integrable directed polymer models

In this paper, we consider four integrable models of directed polymers for which the free energy is known to exhibit KPZ fluctuations. A common framework for the analysis of these models was introduced in our recent work on the O'Connell-Yor polymer. We derive estimates for the central moments of the partition function, of any order, on the near-optimal scale $N^{1/3+ε}$, using an iterative method. Among the innovations exploiting the invariant structure, we develop formulas for correlations between functions of the free energy and the boundary weights that replace the Gaussian integration by parts appearing in the analysis of the O'Connell-Yor case.

math.PR↗

arXiv2020

Central moments of the free energy of the O'Connell-Yor polymer

Seppäläinen and Valkó showed in \cite{SV} that for a suitable choice of parameters, the variance growth of the free energy of the stationary O'Connell-Yor polymer is governed by the exponent $2/3$, characteristic of models in the KPZ universality class. We develop exact formulas based on Gaussian integration by parts to relate the cumulants of the free energy, $\log Z_{n,t}^θ$, to expectations of products of quenched cumulants of the time of the first jump from the boundary into the system, $s_0$. We then use these formulas to obtain estimates for the $k$-th central moment of $\log Z_{n,t}^θ$ as well as the $k$-th annealed moment of $s_0$ for $k> 2$, with nearly optimal exponents $(1/3)k+ε$ and $(2/3)k+ε$, respectively.

math.PR↗

arXiv2017

Fluctuation exponents for stationary exactly solvable lattice polymer models via a Mellin transform framework

We develop a Mellin transform framework which allows us to simultaneously analyze the four known exactly solvable 1+1 dimensional lattice polymer models: the log-gamma, strict-weak, beta, and inverse-beta models. Using this framework we prove the conjectured fluctuation exponents of the free energy and the polymer path for the stationary point-to-point versions of these four models. The fluctuation exponent for the polymer path was previously unproved for the strict-weak, beta, and inverse-beta models.

math.PR↗

arXiv2017

Characterizing stationary 1+1 dimensional lattice polymer models

Motivated by the study of directed polymer models with random weights on the square integer lattice, we define an integrability property shared by the log-gamma, strict-weak, beta, and inverse-beta models. This integrability property encapsulates a preservation in distribution of ratios of partition functions which in turn implies the so called Burke property. We show that under some regularity assumptions, up to trivial modifications, there exist no other models possessing this property.

math.PR↗