Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.4207

Large time asymptotics of growth models on space-like paths II: PNG and parallel TASEP

Alexei Borodin·Patrik L. Ferrari·Tomohiro Sasamoto
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. The joint distributions of surface height at finitely many points at a fixed time moment are given as marginals of a signed determinantal point process. The long time scaling limit of the surface height is shown to coincide with the Airy_1 process. This result holds more generally for the observation points located along any space-like path in the space-time plane. We also obtain the corresponding results for the discrete time TASEP (totally asymmetric simple exclusion process) with parallel update.

Explore related subjects

math-phcond-mat.stat-mechmath.MPmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alexei Borodin, Patrik L. Ferrari, Tomohiro Sasamoto. 2007-07-28. Large time asymptotics of growth models on space-like paths II: PNG and parallel TASEP. https://doi.org/10.1007/s00220-008-0515-4

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

How to Recover Oscillation-Free Pressure in Real Fluids: The RFQC Method and Its Liquid-Upwind Anomaly

From the perspective of continuum thermodynamics, we revisit the pressure oscillation problem in finite-volume methods for multiphase real fluids and clarify the physical counterpart of the Real Fluid Quasi-Conservative (RFQC) method. The pressure oscillation in conservative finite-volume methods originates from their implicit thermodynamic equilibrium assumption, whereas recovering an oscillation-free pressure requires additional physical information. The RFQC method achieves this by evolving the affine parameters xi and E0 of the isentropic internal-energy-pressure relation along pathlines, while the thermodynamic re-projection converts the deviation from the isentropic trajectory into an internal-energy error, thereby ensuring the thermodynamic consistency and numerical stability of the method. We then investigate the applicability limit of the RFQC method and identify a Liquid-upwind Anomaly (LUA) in extreme phase-change cases. For a Riemann problem involving liquid-vapor phase change, a numerical anomaly may occur if a liquid-upwind translational velocity is initially superimposed. Theoretical analysis reveals that this anomaly is initiated by the jump in the affine slope xi during phase change, which delays pressure rise in the downstream vapor cell. Concurrently, the re-projection removes the positive pressure increment, repeatedly generating large internal-energy errors and trapping the vapor cell in a cycle of delayed pressure recovery. The analysis indicates that the LUA is a start-up anomaly, which can be resolved by introducing a regularization strategy at the initial discontinuity. With the proposed regularization strategy, the RFQC method is equipped with enhanced accuracy and robustness for extreme thermodynamic flows, such as sonic phase-change jets.

math-ph↗
arXiv2026

Finite Rodriguez-Villegas Approximants to the Riemann $ξ$-Function

We construct a sequence of finite Rodriguez-Villegas transforms converging locally uniformly to the Riemann $ξ$-function in the critical strip. The input is a positive symmetric profile on the unit interval obtained from the Riemann theta kernel through convolution with the hyperbolic-secant kernel and the logistic coordinate. The profile is a Stieltjes function of $x(1-x)$. Its Bernstein polynomials produce reciprocal numerators and exact finite functional equations. The same numerators admit an exact realization as fermionic supertraces, while the Bernstein polynomials are normalized Gibbs traces.

math-ph↗
arXiv2026

Finite images of braid group representations and algebraic solutions of KZ-type equations

Finite monodromy provides a bridge between group representations and algebraic solutions of differential equations. We study this connection for the Katz-Long-Moody construction, which transforms representations of the semidirect product of a free group and a braid group into new representations of the same group and is related to Knizhnik-Zamolodchikov (KZ)-type equations. For a fixed finite-image input, we classify the parameter values for which the resulting representations have finite image, both on the semidirect product and on its free-group and braid-group subgroups. In particular, finiteness of the braid-group image is independent of the admissible parameter. These results give necessary and sufficient conditions for all solutions of the corresponding regular-singular KZ-type equations to be algebraic. On restriction to the free group, they also characterize finite monodromy and algebraicity of all solutions of the associated Fuchsian systems, connecting the classification to classical questions about algebraic hypergeometric functions.

math-ph↗