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Michael Praehofer

Publications and source records attributed to Michael Praehofer.

9 recordsLinked to original sources
arXiv2023

Domain wall fluctuations of the six-vertex model at the ice point

We report on Monte-Carlo simulations of the six-vertex model with domain wall boundary conditions. In thermal equilibrium such boundary conditions force a fluctuating line separating the disordered region from the perfectly ordered ones. Specifically we study the ice point at which all vertex weights are equal. With high precision the one-point fluctuations of the line are confirmed to be of order $N^{1/3}$ and governed by the Tracy-Widom distribution. Furthermore, the non-universal scaling coefficients are computed for a wide range of interaction strengths. A draft of this paper was completed in January 2019. We improved the presentation and updated references.

cond-mat.stat-mech↗
arXiv2005

One-dimensional stochastic growth and Gaussian ensembles of random matrices

In this review paper we consider the polynuclear growth (PNG) model in one spatial dimension and its relation to random matrix ensembles. For curved and flat growth the scaling functions of the surface fluctuations coincide with limit distribution functions coming from certain Gaussian ensembles of random matrices. This connection can be explained via point processes associated to the PNG model and the random matrices ensemble by an extension to the multilayer PNG and multi-matrix models, respectively. We also explain other models which are equivalent to the PNG model: directed polymers, the longest increasing subsequence problem, Young tableaux, a directed percolation model, kink-antikink gas, and Hammersley process.

math-ph↗
arXiv2004

Exact scaling functions for one-dimensional stationary KPZ growth

We determine the stationary two-point correlation function of the one-dimensional KPZ equation through the scaling limit of a solvable microscopic model, the polynuclear growth model. The equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a Riemann-Hilbert problem related to the Painleve II equation. We solve these equations numerically with very high precision and compare our, up to numerical rounding exact, result with the prediction of Colaiori and Moore [1] obtained from the mode coupling approximation.

cond-mat.stat-mech↗
arXiv2003

Stochastic Growth in One Dimension and Gaussian Multi-Matrix Models

We discuss the space-time determinantal random field which arises for the PNG model in one dimension and resembles the one for Dyson's Brownian motion. The information of interest for growth processes is carried by the edge statistics of the random field and therefore their universal scaling is related to the edge properties of Gaussian multi-matrix models.

math-ph↗
arXiv2003

Fluctuations of an Atomic Ledge Bordering a Crystalline Facet

When a high symmetry facet joins the rounded part of a crystal, the step line density vanishes as sqrt(r) with r denoting the distance from the facet edge. This means that the ledge bordering the facet has a lot of space to meander as caused by thermal activation. We investigate the statistical properties of the border ledge fluctuations. In the scaling regime they turn out to be non-Gaussian and related to the edge statistics of GUE multi-matrix models.

cond-mat.stat-mech↗
arXiv2002

Scale Invariance of the PNG Droplet and the Airy Process

We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process A(y). The Airy process is stationary, it has continuous sample paths, its single "time" (fixed y) distribution is the Tracy-Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has a slow decay of correlations as y^(-2). Roughly the Airy process describes the last line of Dyson's Brownian motion model for random matrices. Our construction uses a multi-layer version of the PNG model, which can be analyzed through fermionic techniques. Specializing our result to a fixed value of y, one reobtains the celebrated result of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation.

math.PR↗
arXiv2000

Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices

We develop a scaling theory for KPZ growth in one dimension by a detailed study of the polynuclear growth (PNG) model. In particular, we identify three universal distributions for shape fluctuations and their dependence on the macroscopic shape. These distribution functions are computed using the partition function of Gaussian random matrices in a cosine potential.

cond-mat.stat-mech↗
arXiv1999

Statistical Self-Similarity of One-Dimensional Growth Processes

For one-dimensional growth processes we consider the distribution of the height above a given point of the substrate and study its scale invariance in the limit of large times. We argue that for self-similar growth from a single seed the universal distribution is the Tracy-Widom distribution from the theory of random matrices and that for growth from a flat substrate it is some other, only numerically determined distribution. In particular, for the polynuclear growth model in the droplet geometry the height maps onto the longest increasing subsequence of a random permutation, from which the height distribution is identified as the Tracy-Widom distribution.

cond-mat.stat-mech↗