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Brownian Gibbs property for Airy line ensembles

- Ivan Corwin, A. Hammond
- Mathematics
- 10 August 2011

Consider a collection of N Brownian bridges $B_{i}:[-N,N] \to \mathbb{R} $, Bi(−N)=Bi(N)=0, 1≤i≤N, conditioned not to intersect. The edge-scaling limit of this system is obtained by taking a weak… Expand

KPZ line ensemble

- Ivan Corwin, A. Hammond
- Mathematics
- 9 December 2013

For each $$t\ge 1$$t≥1 we construct an $$\mathbb {N}$$N-indexed ensemble of random continuous curves with three properties:(1)the lowest indexed curve is distributed as the time t Hopf–Cole solution… Expand

Power law Pólya’s urn and fractional Brownian motion

- A. Hammond, S. Sheffield
- Mathematics
- 6 March 2009

We introduce a natural family of random walks $$S_n$$ on $$\mathbb{Z }$$ that scale to fractional Brownian motion. The increments $$X_n := S_n - S_{n-1} \in \{\pm 1\}$$ have the property that given… Expand

Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation

- A. Hammond
- MathematicsMemoirs of the American Mathematical Society
- 9 September 2016

The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the… Expand

Self-Avoiding Walk is Sub-Ballistic

- H. Duminil-Copin, A. Hammond
- Mathematics
- 2 May 2012

We prove that self-avoiding walk on $${\mathbb{Z}^d}$$Zd is sub-ballistic in any dimension d ≥ 2. That is, writing $${\| u \|}$$‖u‖ for the Euclidean norm of $${u \in \mathbb{Z}^d}$$u∈Zd, and… Expand

The Kinetic Limit of a System of Coagulating Brownian Particles

- A. Hammond, F. Rezakhanlou
- Mathematics
- 29 August 2004

We consider a random model of diffusion and coagulation. A large number of small particles is randomly scattered in $$\mathbb{R}^d$$ at an initial time. Each particle has some integer mass and moves… Expand

Phase Transition for the Speed of the Biased Random Walk on the Supercritical Percolation Cluster

- A. Fribergh, A. Hammond
- Mathematics
- 7 March 2011

We prove the sharpness of the phase transition for the speed in biased random walk on the supercritical percolation cluster on ℤd. That is, for each d ≥ 2, and for any supercritical parameter p > pc,… Expand

A PATCHWORK QUILT SEWN FROM BROWNIAN FABRIC: REGULARITY OF POLYMER WEIGHT PROFILES IN BROWNIAN LAST PASSAGE PERCOLATION

- A. Hammond
- MathematicsForum of Mathematics, Pi
- 13 September 2017

In last passage percolation models lying in the Kardar–Parisi–Zhang (KPZ) universality class, the energy of long energy-maximizing paths may be studied as a function of the paths’ pair of endpoint… Expand

Biased random walks on a Galton-Watson tree with leaves

- G. B. Arous, A. Fribergh, N. Gantert, A. Hammond
- Mathematics
- 23 November 2007

We consider a biased random walk $X_n$ on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant $\gamma= \gamma(\beta) \in (0,1)$, depending on… Expand

Fractal geometry of Airy$_{2}$ processes coupled via the Airy sheet

- Riddhipratim Basu, S. Ganguly, A. Hammond
- Mathematics
- 3 April 2019

In last passage percolation models lying in the Kardar-Parisi-Zhang universality class, maximizing paths that travel over distances of order $n$ accrue energy that fluctuates on scale $n^{1/3}$; and… Expand

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