Universal fluctuation of one-dimensional KPZ growth.pdf

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Universal fluctuation of one-dimensional KPZ growth.pdf

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Universal fluctuation of one-dimensional KPZ growth

J.S tat.M ech. (2006) P 08014 ournal of Statistical Mechanics: An IOP and SISSA journalJ Theory and Experiment Universal fluctuation of the average height in the early-time regime of one-dimensional Kardar–Parisi– Zhang-type growth Deok-Sun Lee1 and Doochul Kim2 1 Theoretische Physik, Universita?t des Saarlandes, 66041 Saarbru?cken, Germany 2 School of Physics and Astronomy and Centre for Theoretical Physics, Seoul National University, 151-747, Korea E-mail: deoksun.lee@ and dkim@snu.ac.kr Received 25 May 2006 Accepted 26 July 2006 Published 21 August 2006 Online at /JSTAT/2006/P08014 doi:10.1088/1742-5468/2006/08/P08014 Abstract. The statistics of the average height fluctuation of the one- dimensional Kardar–Parisi–Zhang (KPZ)-type surface is investigated. Guided by the idea of local stationarity, we derive the scaling form of the characteristic function in the early-time regime, t N3/2 with t time and N the system size, from the known characteristic function in the stationary state (t N3/2) of the single-step model derivable from a Bethe ansatz solution, and thereby find the scaling properties of the cumulants and the large deviation function in the early-time regime. These results, combined with the scaling analysis of the KPZ equation, imply the existence of the universal scaling functions for the cumulants and an universal large deviation function. The analytic predictions are supported by the simulation results for three different models in the KPZ class. Keywords: integrable spin chains (vertex models), kinetic growth processes (theory), fluctuations (theory) c?2006 IOP Publishing Ltd and SISSA 1742-5468/06/P08014+11$30.00 J.S tat.M ech. (2006) P 08014 Universal fluctuation of one-dimensional KPZ growth Contents 1. Introduction 2 2. LDF in the stationary state of the SS model 3 3. LDF in the early-time regime of the SS model 5 3.1. Local stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2. Dynamic scaling and LDF in the earl