The Annals of Applied Probability

Scaling limit for Brownian motions with one-sided collisions

Patrik L. Ferrari, Herbert Spohn, and Thomas Weiss

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Abstract

We consider Brownian motions with one-sided collisions, meaning that each particle is reflected at its right neighbour. For a finite number of particles a Schütz-type formula is derived for the transition probability. We investigate an infinite system with periodic initial configuration, that is, particles are located at the integer lattice at time zero. The joint distribution of the positions of a finite subset of particles is expressed as a Fredholm determinant with a kernel defining a signed determinantal point process. In the appropriate large time scaling limit, the fluctuations in the particle positions are described by the $\mathrm{Airy}_{1}$ process.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 3 (2015), 1349-1382.

Dates
First available in Project Euclid: 23 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1427124131

Digital Object Identifier
doi:10.1214/14-AAP1025

Mathematical Reviews number (MathSciNet)
MR3325276

Zentralblatt MATH identifier
1315.60108

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J65: Brownian motion [See also 58J65]

Keywords
Brownian motion one-sided collision Airy$_{1}$ process Fredholm determinant periodic initial configuration

Citation

Ferrari, Patrik L.; Spohn, Herbert; Weiss, Thomas. Scaling limit for Brownian motions with one-sided collisions. Ann. Appl. Probab. 25 (2015), no. 3, 1349--1382. doi:10.1214/14-AAP1025. https://projecteuclid.org/euclid.aoap/1427124131


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References

  • [1] Anderson, R. F. and Orey, S. (1976). Small random perturbation of dynamical systems with reflecting boundary. Nagoya Math. J. 60 189–216.
  • [2] Banwell, T. C. and Jayakumar, A. (2000). Exact analytical solution for current flow through diode with series resistance. Electronics Letters 36 291–292.
  • [3] Barry, D. A., Parlange, J.-Y., Li, L., Prommer, H., Cunningham, C. J. and Stagnitti, F. (2000). Analytical approximations for real values of the Lambert $W$-function. Math. Comput. Simulation 53 95–103.
  • [4] Baryshnikov, Yu. (2001). GUEs and queues. Probab. Theory Related Fields 119 256–274.
  • [5] Ben Arous, G. and Corwin, I. (2011). Current fluctuations for TASEP: A proof of the Prähofer–Spohn conjecture. Ann. Probab. 39 104–138.
  • [6] Borodin, A. and Corwin, I. (2014). Macdonald processes. Probab. Theory Related Fields 158 225–400.
  • [7] Borodin, A. and Ferrari, P. L. (2008). Large time asymptotics of growth models on space-like paths. I. PushASEP. Electron. J. Probab. 13 1380–1418.
  • [8] Borodin, A., Ferrari, P. L. and Prähofer, M. (2007). Fluctuations in the discrete TASEP with periodic initial configurations and the Airy$_{1}$ process. Int. Math. Res. Papers 2007 rpm002.
  • [9] Borodin, A., Ferrari, P. L., Prähofer, M. and Sasamoto, T. (2007). Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129 1055–1080.
  • [10] Borodin, A., Ferrari, P. L. and Sasamoto, T. (2008). Transition between ${\mathrm{Airy}}_{1}$ and ${\mathrm{Airy}}_{2}$ processes and TASEP fluctuations. Comm. Pure Appl. Math. 61 1603–1629.
  • [11] Chang, C. C. and Yau, H.-T. (1992). Fluctuations of one-dimensional Ginzburg–Landau models in nonequilibrium. Comm. Math. Phys. 145 209–234.
  • [12] Chen, Y. and Moore, K. L. (2002). Analytical stability bound for delayed second-order systems with repeating poles using Lambert function $W$. Automatica J. IFAC 38 891–895.
  • [13] Corless, R. M., Gonnet, G. H., Hare, D. E. G. and Jeffrey, D. J. (1993). Lambert’s W function in Maple. The Maple Technical Newsletter 9 12–22.
  • [14] Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. and Knuth, D. E. (1996). On the Lambert W function. Adv. Comput. Math. 5 329–359.
  • [15] Corless, R. M., Jeffrey, D. J. and Knuth, D. E. (1997). A sequence of series for the Lambert $W$ function. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Kihei, HI) 197–204 (electronic). ACM, New York.
  • [16] Corless, R. M., Jeffrey, D. J. and Valluri, S. R. (2000). Some applications of the Lambert W function to physics. Canadian Journal of Physics 78 823–831.
  • [17] Corwin, I., Ferrari, P. L. and Péché, S. (2012). Universality of slow decorrelation in KPZ growth. Ann. Inst. Henri Poincaré Probab. Stat. 48 134–150.
  • [18] Ferrari, P. L. (2008). Slow decorrelations in KPZ growth. J. Stat. Mech. 2008 P07022.
  • [19] Forrester, P. J. and Nagao, T. (2011). Determinantal correlations for classical projection processes. J. Stat. Mech. 2011 P08011.
  • [20] Harris, T. E. (1965). Diffusion with “collisions” between particles. J. Appl. Probab. 2 323–338.
  • [21] Harrison, J. M. and Williams, R. J. (1987). Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab. 15 115–137.
  • [22] Jain, A. and Kapoor, A. (2004). Exact analytical solutions of the parameters of real solar cells using Lambert W-function. Solar Energy Materials and Solar Cells 81 269–277.
  • [23] Karatzas, I., Pal, S. and Shkolnikov, M. (2012). Systems of Brownian particles with asymmetric collisions. Available at arXiv:1210.0259.
  • [24] Kardar, M., Parisi, G. and Zhang, Y. Z. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 889–892.
  • [25] Ledoux, M. (2007). Deviation inequalities on largest eigenvalues. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1910 167–219. Springer, Berlin.
  • [26] O’Connell, N. (2012). Directed polymers and the quantum Toda lattice. Ann. Probab. 40 437–458.
  • [27] O’Connell, N. and Yor, M. (2001). Brownian analogues of Burke’s theorem. Stochastic Process. Appl. 96 285–304.
  • [28] Sasamoto, T. (2005). Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38 L549–L556.
  • [29] Sasamoto, T. and Wadati, M. (1998). Determinant form solution for the derivative nonlinear Schrödinger type model. J. Phys. Soc. Japan 67 784–790.
  • [30] Schütz, G. M. (1997). Exact solution of the master equation for the asymmetric exclusion process. J. Stat. Phys. 88 427–445.
  • [31] Skorokhod, A. V. (1961). Stochastic equations for diffusions in a bounded region. Theory Probab. Appl. 6 264–274.
  • [32] Tracy, C. A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151–174.
  • [33] Tracy, C. A. and Widom, H. (1996). On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 727–754.
  • [34] Varadhan, S. R. S. and Williams, R. J. (1985). Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math. 38 405–443.
  • [35] Warren, J. (2007). Dyson’s Brownian motions, intertwining and interlacing. Electron. J. Probab. 12 573–590.
  • [36] Weiss, T. (2011). Scaling behaviour of the directed polymer model of Baryshnikov and O’Connell at zero temperature. Bachelor thesis, TU-München.