Abstract
The Airy distribution function describes the probability distribution of the area under a Brownian excursion over a unit interval. Surprisingly, this function has appeared in a number of seemingly unrelated problems, mostly in computer science and graph theory. In this paper, we show that this distribution function also appears in a rather well studied physical system, namely the fluctuating interfaces. We present an exact solution for the distribution P(h m ,L) of the maximal height h m (measured with respect to the average spatial height) in the steady state of a fluctuating interface in a one dimensional system of size L with both periodic and free boundary conditions. For the periodic case, we show that P(h m ,L)=L−1/2f(h m L−1/2) for all L>0 where the function f(x) is the Airy distribution function. This result is valid for both the Edwards–Wilkinson (EW) and the Kardar–Parisi–Zhang interfaces. For the free boundary case, the same scaling holds P(h m ,L)=L−1/2F(h m L−1/2), but the scaling function F(x) is different from that of the periodic case. We compute this scaling function explicitly for the EW interface and call it the F-Airy distribution function. Numerical simulations are in excellent agreement with our analytical results. Our results provide a rather rare exactly solvable case for the distribution of extremum of a set of strongly correlated random variables. Some of these results were announced in a recent Letter [S.N. Majumdar and A. Comtet, Phys. Rev. Lett. 92: 225501 (2004)].
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References
D. A. Darling (1983) ArticleTitleOn the supremum of certain Gaussian processes Ann. Prob. 11 803–806
G. Louchard (1984) ArticleTitleKac’s formula, Levy’s local time and Brownian excursion J. Appl. Prob. 21 479–499
L. Takacs, A Bernoulli excursion and its various applications, Adv. Appl. Prob. 23:557–585, (1991); Limit distributions for the Bernoulli meander, J. Appl. Prob. 32:375–395 (1995).
M. Abramowitz I. A. Stegun (1973) Handbook of Mathematical Functions Dover New York
M. Csörgö Z. Shi M. Yor (1999) ArticleTitleSome asymptotic properties of the local time of the uniform empirical processes Bernoulli 5 1035–1058
P. Flajolet P. Poblete A. Viola (1998) ArticleTitleOn the analysis of linear probing hashing Algorithmica 22 490–515
P. Flajolet G. Louchard (2001) ArticleTitleAnalytic variations on the Airy distribution Algorithmica 31 361–377 Occurrence Handle10.1007/s00453-001-0056-0
P. Flajolet B. Salvy G. Schaeffer (2004) ArticleTitleAiry phenomena and analytic combinatorics of connected graphs Electron. J. Combin. 11 1–30
C. L. Mallows and J. Riordan, The inversion enumerator for labelled trees, Bull. Am. Math. Soc. 74:92–94 (1968); I. Gessel, B. E. Sagan, and Y.-N. Yeh, Enumeration of trees by inversion, J. Graph Theory 19:435–459 (1995).
E. M. Wright, The number of connected sparsely edged graphs, J. Graph Theory 1:317–330 (1977); 2- Smooth graphs and blocks 2:299–305 (1978); 3-Asymptotic results 4:393–407 (1980).
P. Flajolet, D. E. Knuth, and B. Pittel, The first cycles in an evolving graph, Discrete Math. 75:167–215 (1989); S. Janson, D. E. Knuth, T. Luczak, and B. Pittel, Random Struct and Algorithms 4:233 (1993).
C. Richard, A. J. Guttmann, and I. Jensen, Scaling function and universal amplitude combinations for self avoiding polygons, J. Phys. A: Math. Gen. 34:L495–501 (2001); C. Richard, Scaling behaviour of the two-dimensional polygons models, J. Stat. Phys. 108:459–493 (2002); C. Richard, I. Jensen, and A. J. Guttmann, Scaling function for self-avoiding polygons, Proceedings of the International Congress on Theoretical Physics, Paris, July 2002, D. Iagolnitzer, D. Rivasseau, and J. Zinn-Justin, eds. (Birkhauser), cond-mat/0302513.
C. Richard (2004) ArticleTitleArea distribution of the planar random loop boundary J. Phys. A: Math. Gen. 37 4493–4500 Occurrence Handle10.1088/0305-4470/37/16/002 Occurrence HandleMR2065940
S.N. Majumdar A. Comtet (2004) ArticleTitleExact maximal height distribution of fluctuating interfaces Phys. Rev. Lett. 92 225501 Occurrence Handle10.1103/PhysRevLett.92.225501 Occurrence Handle15245233
A. L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, England, 1995); J. Krug, Adv. Phys. 46:139–282 (1997).
T. Halpin-Healy Y.-C. Zhang (1995) ArticleTitleKinetic roughening phenomena, stochastic growth, directed polymers and all that Phys. Rep. 254 215–414 Occurrence Handle10.1016/0370-1573(94)00087-J
E. J. Gumbel (1958) Statistics of Extremes Columbia University Press New York
J.-P. Bouchaud and M. Mézard, Universality classes for extreme value statistics, J. Phys. A 30:7997–8015 (1997); D. Carpentier and P. Le Doussal, Glass transition of a particle in a random potential, front selection in non-linear renormalization group and entropic phenomena in Liouville and sinh-Gordon models, Phys. Rev. E 63:026110 (2001); D. S. Dean and S. N. Majumdar, Extreme-value statistics of hierarchically correlated variables, deviation from Gumbel statistics and anomalous persistence, Phys. Rev. E 64:046121 (2001); P. LeDoussal and C. Monthus, Exact solutions for the statistics of extrema of some random 1D landscapes, application to the equilibrium and the dynamics of the toy model, Physica A 317:140–198 (2003).
S.N. Majumdar P.L. Krapivsky (2003) ArticleTitleExtreme value statistics and traveling fronts: various applications Physica A 318 161–170
For a review see, E. Ben-Naim, P. L. Krapivsky, and S. Redner, extremal properties of random structures, cond-mat/0311552.
S. Raychaudhuri M. Cranston C. Przybyla Y. Shapir (2001) ArticleTitleMaximal height scaling of kinetically growing surfaces Phys. Rev. Lett. 87 136101 Occurrence Handle10.1103/PhysRevLett.87.136101 Occurrence Handle1:STN:280:DC%2BD3MrjtlyjtQ%3D%3D Occurrence Handle11580607
S. F. Edwards D. R. Wilkinson (1982) ArticleTitleThe surface statistics of a granular agregate Proc. R. Soc. London A 381 17–31
M. Kardar G. Parisi Y.-C. Zhang (1986) ArticleTitleDynamical scaling of growing interfaces Phys. Rev. Lett. 56 889–892 Occurrence Handle10.1103/PhysRevLett.56.889 Occurrence Handle1:CAS:528:DyaL28XitlWhs7w%3D Occurrence Handle10033312
M. Perman J. A. Wellner (1996) ArticleTitleOn the distribution of Brownian areas Ann. Appl. Prob. 6 1091–1111 Occurrence Handle10.1214/aoap/1035463325
M. Jeanblanc J. Pitman M. Yor (1997) ArticleTitleThe Feynman–Kac formula and decomposition of Brownian paths Comput. Appl. Math. 16 27–52
M. Nguyen Thê, Area and inertial moments of Dyck paths, Combinatorics, Probability and Computing (2003), submitted.
S. Redner (2001) A Guide to First-passage Processes Cambridge University Press Cambridge
J. R. Albright (1977) ArticleTitleIntegrals of products of Airy functions J. Phys. A 10 485–490
G. Foltin, K. Oerding, Z. Racz, R. L. Workman, and R. K. P. Zia, Width distribution for random-walk interfaces, Phys. Rev. E 50:R639–642 (1994); Z. Racz and M. Plischke, Width distribution for 2+1 dimensional growth and deposition processes, Phys. Rev. E 50:3530–3537 (1994).
T. W. Burkhardt (1993) ArticleTitleSemiflexible polymer in the half plane and statistics of the integral of a Brownian curve J. Phys. A 26 L1157–1162 Occurrence Handle1:CAS:528:DyaK2cXmsFKkt7o%3D
A. Yu. Grosberg and A. R. Khokhlov, Statistical Physics of Macromolecules (AIP Press, 1994).
W. Feller (1968) Introduction to Probability Theory and its Applications EditionNumber3 Wiley New York
J. Amar F. Family (1990) ArticleTitleNumerical solution of continuum equation for interface growth in 2+1 dimension Phys. Rev. A 41 3399–3402 Occurrence Handle10.1103/PhysRevA.41.3399 Occurrence Handle9903504
K. Moser, J. Kertész, and D. E. Wolf, Numerical solution of the Kardar–Parisi–Zhang equation in one,two and three dimensions, Physica A 178:215–226 (1991); K. Moser and D. E. Wolf, Vectorized and parallel simulations of the Kardar–Parisi–Zhang equation in 3+1 dimensions, J. Phys. A 27:4049–4054 (1994).
C. Dasgupta, S. Das Sarma, and J. M. Kim, Controlled instability and multiscaling in models of epitaxial growth, Phys. Rev. E 54:R4552–4555 (1996); C. Dasgupta, J. M. Kim, M. Dutta, and S. Das Sarma, Instability, intermittency, and multiscaling in discrete growth models of kinetic roughening, ibid. 55:2235–2254 (1997).
T. J. Newman A. J. Bray (1996) ArticleTitleStrong coupling behaviour in discrete Kardar–Parisi–Zhang equation J. Phys. A 29 7917–7928
C.-H. Lam F. G. Shin (1998) ArticleTitleAnomaly in numerical integration of the Kardar–Parisi–Zhang equation Phys. Rev. E 57 6506–6511 Occurrence Handle10.1103/PhysRevE.57.6506 Occurrence Handle1:CAS:528:DyaK1cXjvVKnsbk%3D
C.-H. Lam F. G. Shin (1998) ArticleTitleImproved discretisation of the Kardar–Parisi–Zhang equation Phys. Rev. E 58 5592–5595 Occurrence Handle10.1103/PhysRevE.58.5592 Occurrence Handle1:CAS:528:DyaK1cXntFGmsb8%3D
See Eq. (5.26) in ref. 40 where this result was attributed to unpublished work (1989) of T. Nieuwenhuizen.
For a review, see J. Krug and H. Spohn, in Solids Far From Equilibrium: Growth, Morphology and Defects, C. Godreche, ed. (Cambridge University Press, Cambridge, 1991).
P. Meakin et al. (1986) ArticleTitleBallistic deposition on surfaces Phys. Rev. A 34 5091 Occurrence Handle10.1103/PhysRevA.34.5091 Occurrence Handle9897896
B. Derrida et al. (1993) ArticleTitleExact solution of a 1-d asymmetric exclusion model using a matrix formulation J. Phys. A 26 1493
For a review, see B. Derrida and M. R. Evans in Nonequilibrium Statistical Mechanics in One Dimension, V. Privman, ed. (Cambridge University Press, Cambridge, 1997).
B. Derrida and J. L. Lebowitz, Exact large deviation function in the asymmetric exclusion process, Phys. Rev. Lett. 80:209–213 (1998); B. Derrida and C. Appert, Universal large deviation function of the Kardar–Parisi–Zhang equation in one dimension, J. Stat. Phys. 94:1–30 (1999).
M. Praehofer and H. Spohn, Universal distribution of growth processes in 1+1 dimensions and random matrices, Phys. Rev. Lett. 84:4882–4885 (2000); Exact scaling function for one-dimensional stationary KPZ growth, J. Stat. Phys. 115:255–279 (2002). K. Johansson, Shape fluctuations and random matrices Commun. Math. Phys. 209:437–476 (2000); J. Gravner, C. A. Tracy and H. Widom, Limit theorems for height fluctuations in a class of discrete space and time growth models J. Stat. Phys. 102:1085–1132 (2001); S. N. Majumdar and S. Nechaev, Anisotropic ballistic deposition model with links to the Ulam problem and the Tracy–Widom distribution, Phys. Rev. E 69:011103 (2003); P. L. Ferrari, Ploynuclear growth on a flat substrate and edge scaling of GOE eigenvalues, Comm. Math. Phys. 252:77–109 (2004); T. Imamura and T. Sasamoto, Fluctuations of a one-dimensional polynuclear growth model in a half space, J. Stat. Phys. 115:749–803 (2004).
C. A. Tracy H. Widom (1994) ArticleTitleLevel-spacing distributions and the Airy kernel Commun. Math. Phys. 159 151–174
G. Gyorgyi et al. (2003) ArticleTitleStatistics of extreme intensities for Gaussian interfaces Phys. Rev. E 68 056116 Occurrence Handle10.1103/PhysRevE.68.056116 Occurrence Handle1:STN:280:DC%2BD2c%2FjsF2gtA%3D%3D
D. B. Dougherty et al., Experimental persistence probability for fluctuating steps, Phys. Rev. Lett. 89:136102–136106 (2002); M. Constantin et al., Infinite family of persistence exponents for interface fluctuations, Phys. Rev. Lett. 91:086103 (2003); C. Dasgupta et al., Survival in equilibrium step fluctuations, Phys. Rev. E 69:022101 (2004).
M. Giesen (2001) ArticleTitleStep and island dynamics at solid/vacuum and solid/liquid interfaces Prog. Surf. Sci. 68 1–153 Occurrence Handle10.1016/S0079-6816(00)00021-6 Occurrence Handle1:CAS:528:DC%2BD3MXkvVeru78%3D
R. Toussaint, G. Helgesen, and E. G. Flekkoy, Dynamic roughening and fluctuations of dipolar chains, cond-mat/0311340.
J. Krug et al., Persistence exponents for fluctuating interfaces, Phys. Rev. E 56:2702–2712 (1997); H. Kallabis and J. Krug, Persistence of Kardar–Parisi–Zhang interfaces Europhys. Lett. 45:20–25 (1999); Z. Toroczkai and E. D. Williams, Nanoscale fluctuations at solid interfaces, Phys. Today 52:No. 12,24–28 (1999); S. N. Majumdar and A. J. Bray, Spatial persistence of fluctuating interfaces, Phys. Rev. Lett. 86:3700–3703 (2001). J. Krug, Power law in surface physics: the deep, the shallow and the useful, Physica A 340:647 (2004).
H. Guclu G. Korniss (2004) ArticleTitleExtreme fluctuations in small-world networks with relaxional dynamics Phys. Rev. E 69 065104(R) Occurrence Handle10.1103/PhysRevE.69.065104
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Majumdar, S.N., Comtet, A. Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces. J Stat Phys 119, 777–826 (2005). https://doi.org/10.1007/s10955-005-3022-4
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DOI: https://doi.org/10.1007/s10955-005-3022-4