# 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} }

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…

## 22 Citations

Asymptotic Behaviour of Semi-Infinite Geodesics for Maximal Increasing Subsequences in the Plane

- Mathematics
- 2002

We consider for a given Poissonian cloud ω in ℝ2the maximal up/right path fromxto y(x, y∈ ℝ2), where maximal means that it contains as many points ofω as possible. We prove that with probability 1…

Superdiffusivity for Brownian Motion in a Poissonian Potential with Long Range Correlation I: Lower Bound on the Volume Exponent

- Mathematics, Physics
- 2011

This paper continues a study on trajectories of Brownian Motion in a field of soft trap whose radius distribution is unbounded. We show here for both point-to-point and point-to-plane model the…

Numerical bounds for critical exponents of crossing Brownian motion

- Mathematics
- 2001

We consider d-dimensional crossing Brownian motion in a truncated Poissonian potential conditioned to reach a fixed hyperplane at distance L from the starting point. The transverse fluctuation of the…

Transversal Fluctuations of the ASEP, Stochastic Six Vertex Model, and Hall-Littlewood Gibbsian Line Ensembles

- Mathematics
- 2017

We consider the ASEP and the stochastic six vertex model started with step initial data. After a long time, T, it is known that the one-point height function fluctuations for these systems are of…

A Universality Property for Last-Passage Percolation Paths Close to the Axis

- Mathematics
- 2004

We consider a last-passage directed percolation model in $Z_+^2$, with i.i.d. weights whose common distribution has a finite $(2+p)$th moment. We study the fluctuations of the passage time from the…

Longest increasing path within the critical strip

- Mathematics
- 2018

A Poisson point process of unit intensity is placed in the square $[0,n]^2$. An increasing path is a curve connecting $(0,0)$ with $(n,n)$ which is non-decreasing in each coordinate. Its length is…

Two Results on Asymptotic Behaviour of Random Walks in Random Environment

- Mathematics
- 2016

Two results on Asymptotic Behaviour of Random Walks in Random Environment Jeremy Voltz Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2016 In the first chapter of this…

Convergence of the Environment Seen from Geodesics in Exponential Last-Passage Percolation

- Mathematics
- 2021

A well-known question in the planar first-passage percolation model concerns the convergence of the empirical distribution along geodesics. We demonstrate this convergence for an explicit model,…

Small deviation estimates and small ball probabilities for geodesics in last passage percolation

- Mathematics
- 2021

For the exactly solvable model of exponential last passage percolation on Z, consider the geodesic Γn joining (0, 0) and (n, n) for large n. It is well known that the transversal fluctuation of Γn…

Nonexistence of Bigeodesics in Integrable Models of Last Passage Percolation

- Mathematics, Physics
- 2018

Bi-infinite geodesics are fundamental objects of interest in planar first passage percolation. A longstanding conjecture states that under mild conditions there are almost surely no bigeodesics,…

## References

SHOWING 1-10 OF 20 REFERENCES

Large deviations for increasing sequences on the plane

- Mathematics
- 1998

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

- Mathematics
- 1996

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

- Mathematics
- 1999

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…

Hammersley's interacting particle process and longest increasing subsequences

- Mathematics
- 1995

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

- Mathematics
- 1998

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

- Mathematics
- 1999

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

- Education
- 1972

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

- Physics, Mathematics
- 1998

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

- Mathematics
- 1995

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…