Transversal fluctuations for increasing subsequences on the plane

@article{Johansson1999TransversalFF,
  title={Transversal fluctuations for increasing subsequences on the plane},
  author={Kurt Johansson},
  journal={Probability Theory and Related Fields},
  year={1999},
  volume={116},
  pages={445-456}
}
  • K. Johansson
  • Published 27 October 1999
  • Mathematics
  • Probability Theory and Related Fields
Abstract. Consider a realization of a Poisson process in ℝ2 with intensity 1 and take a maximal up/right path from the origin to (N, N) consisting of line segments between the points, where maximal means that it contains as many points as possible. The number of points in such a path has fluctuations of order Nχ, where χ = 1/3, [BDJ]. Here we show that typical deviations of a maximal path from the diagonal x = y is of order Nξ with ξ = 2/3. This is consistent with the scaling identity χ = 2ξ− 1… 
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References

SHOWING 1-10 OF 20 REFERENCES
Large deviations for increasing sequences on the plane
Abstract. We prove a large deviation principle with explicit rate functions for the length of the longest increasing sequence among Poisson points on the plane. The rate function for lower tail…
Superdiffusivity in first-passage percolation
Summary.In standard first-passage percolation on ${\Bbb Z}^d$ (with $d\geq 2$), the time-minimizing paths from a point to a plane at distance $L$ are expected to have transverse fluctuations of order…
Algebraic aspects of increasing subsequences
We present a number of results relating partial Cauchy-Littlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for…
Level-spacing distributions and the Airy kernel
Hammersley's interacting particle process and longest increasing subsequences
SummaryIn a famous paper [8] Hammersley investigated the lengthLn of the longest increasing subsequence of a randomn-permutation. Implicit in that paper is a certain one-dimensional continuous-space…
On the distribution of the length of the longest increasing subsequence of random permutations
Let SN be the group of permutations of 1,2,..., N. If 7r E SN, we say that 7(i1),... , 7F(ik) is an increasing subsequence in 7r if il < i2 < ... < ik and 7r(ii) < 7r(i2) < ...< 7r(ik). Let 1N(r) be…
Shape Fluctuations and Random Matrices
Abstract: We study a certain random growth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations,…
A few seedlings of research
Graduate students sometimes ask. or fail to ask: "How does one do research in mathematical statistics? It is a reasonable question because the fruits of research, lectures and published papers bear…
Scaling identity for crossing Brownian motion in a Poissonian potential
Abstract. We consider d-dimensional Brownian motion in a truncated Poissonian potential (d≥ 2). If Brownian motion starts at the origin and ends in the closed ball with center y and radius 1, then…
Divergence of Shape Fluctuations in Two Dimensions
We consider stochastic growth models, such as standard first-passage percolation on Z d , where to leading order there is a linearly growing deterministic shape. Under natural hypotheses, we prove…
...
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