Skip to main content

Packed and Periodic Initial Conditions

  • Chapter
  • First Online:
Reflected Brownian Motions in the KPZ Universality Class

Part of the book series: SpringerBriefs in Mathematical Physics ((BRIEFSMAPHY,volume 18))

  • 679 Accesses

Abstract

Packed initial conditions turn out to be the most readily accessible case. Although such initial conditions have been studied extensively in the literature, we provide here a complete proof, introducing essentially all methods required later on in more complicated situations. The second treatable case of deterministic initial data are periodic initial conditions. Here we will be rather brief, in stating only the main results, since the tools used in Ferrari et al. (2015) (Ferrari et al., Ann. Appl. Probab. 25, 1349–1382, 2015) are comparable to the ones employed in case of packed initial conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • P.L. Ferrari, H. Spohn, T. Weiss, Scaling limit for Brownian motions with one-sided collisions. Ann. Appl. Probab. 25, 1349–1382 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • J. Warren, Dyson’s Brownian motions, intertwining and interlacing. Electron. J. Probab. 12, 573–590 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • A. Borodin, P.L. Ferrari, M. Prähofer, T. Sasamoto, Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129, 1055–1080 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • J. Gravner, C.A. Tracy, H. Widom, Limit theorems for height fluctuations in a class of discrete space and time growth models. J. Stat. Phys. 102, 1085–1132 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Y. Chen, K.L. Moore, Analytical stability bound for delayed second-order systems with repeating poles using Lambert function W. Automatica 38, 891–895 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • T.C. Banwell, A. Jayakumar, Exact analytical solution for current flow through diode with series resistance. Electron. Lett. 36, 291–292 (2000)

    Article  Google Scholar 

  • A. Jain, A. Kapoor, Exact analytical solutions of the parameters of real solar cells using Lambert W-function. Solar Energy Mater. Solar Cells 81, 269–277 (2004)

    Article  Google Scholar 

  • R.M. Corless, D.J. Jeffrey, S.R. Valluri, Some applications of the Lambert W function to physics. Canadian J. Phys. 78, 823–831 (2000)

    ADS  Google Scholar 

  • D.A. Barry, J.-Y. Parlange, L. Li, H. Prommer, C.J. Cunningham, F. Stagnitti, Analytical approximations for real values of the Lambert W-function. Math. Comput. Simul. 53, 95–103 (2000)

    Article  MathSciNet  Google Scholar 

  • R.M. Corless, D.J. Jeffrey, D.E. Knuth, A sequence of series for the Lambert W function, in Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (1997), pp. 197–204

    Google Scholar 

  • R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, Lambert’s W function in maple. Maple Tech. Newsl. 9, 12–22 (1993)

    Google Scholar 

  • R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, D.E. Knuth, On the Lambert W function. Adv. Comput. Math. 5, 329–359 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Herbert Spohn .

Rights and permissions

Reprints and permissions

Copyright information

© 2017 The Author(s)

About this chapter

Cite this chapter

Weiss, T., Ferrari, P., Spohn, H. (2017). Packed and Periodic Initial Conditions. In: Reflected Brownian Motions in the KPZ Universality Class. SpringerBriefs in Mathematical Physics, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-319-49499-9_5

Download citation

Publish with us

Policies and ethics