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 Electronic Communications in Probability > Vol. 15(2010) > Paper 1 open journal systems 


Soft edge results for longest increasing paths on the planar lattice

Nicos Georgiou, UW-Madison


Abstract
For two-dimensional last-passage time models of weakly increasing paths, interesting scaling limits have been proved for points close the axis (the hard edge). For strictly increasing paths of Bernoulli($p$) marked sites, the relevant boundary is the line $y=px$. We call this the soft edge to contrast it with the hard edge. We prove laws of large numbers for the maximal cardinality of a strictly increasing path in the rectangle $[p^{-1}n -xn^a]times[n]$ as the parameters $a$ and $x$ vary. The results change qualitatively as $a$ passes through the value $1/2$.


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Pages: 1-13

Published on: January 7, 2010
Bibliography
  1. Baik, Jinho; Suidan, Toufic M. A GUE central limit theorem and universality of directed first and last passage site percolation. Int. Math. Res. Not. 2005, no. 6, 325--337. Math. Review 2006c:60025
  2. Bodineau, Thierry; Martin, James. A universality property for last-passage percolation paths close to the axis. Electron. Comm. Probab. 10 (2005), 105--112 (electronic). Math. Review 2006a:60189
  3. Cohn, Henry; Elkies, Noam; Propp, James. Local statistics for random domino tilings of the Aztec diamond. Duke Math. J. 85 (1996), no. 1, 117--166. Math. Review 97k:52026
  4. Dieker, A. B.; Warren, J. Determinantal transition kernels for some interacting particles on the line. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), no. 6, 1162--1172. Math. Review 2010a:60236
  5. Glynn, Peter W.; Whitt, Ward. Departures from many queues in series. Ann. Appl. Probab. 1 (1991), no. 4, 546--572.Math. Review 92i:60162
  6. William Jockusch, James Propp, and Peter Shor. Random domino tilings and the arctic circle theorem. arXiv:math/9801068. Math Review number is not available.
  7. Kallenberg, Olav. Foundations of modern probability. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2 Math. Review 2002m:60002
  8. Majumdar, Satya N.; Mallick, Kirone; Nechaev, Sergei. Bethe ansatz in the Bernoulli matching model of random sequence alignment. Phys. Rev. E (3) 77 (2008), no. 1, 011110, 10 pp. Math. Review 2009j:82015
  9. V.B. Priezzhev and G.M. Schütz. Exact solution of the Bernoulli matching model of sequence alignment. J. Stat. Mech.}, 2008, P09007. The Math. Review number is not available.
  10. Rákos, A.; Schütz, G. M. Current distribution and random matrix ensembles for an integrable asymmetric fragmentation process. J. Stat. Phys. 118 (2005), no. 3-4, 511--530. Math. Review 2005k:82087
  11. Rost, H. Nonequilibrium behaviour of a many particle process: density profile and local equilibria. Z. Wahrsch. Verw. Gebiete 58 (1981), no. 1, 41--53. Math. Review 83a:60176
  12. Seppäläinen, Timo. Increasing sequences of independent points on the planar lattice. Ann. Appl. Probab. 7 (1997), no. 4, 886--898. Math. Review 99a:60116
  13. Seppäläinen, Timo. A scaling limit for queues in series. Ann. Appl. Probab. 7 (1997), no. 4, 855--872. Math. Review 99j:60160
  14. Seppäläinen, T. Hydrodynamic scaling, convex duality and asymptotic shapes of growth models. Markov Process. Related Fields 4 (1998), no. 1, 1--26. Math. Review 99e:60221
















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Electronic Communications in Probability. ISSN: 1083-589X