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Brownian Gibbs property for Airy line ensembles
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…
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KPZ line ensemble
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…
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Power law Pólya’s urn and fractional Brownian motion
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…
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Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation
  • A. Hammond
  • Mathematics
    Memoirs 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…
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Self-Avoiding Walk is Sub-Ballistic
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…
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The Kinetic Limit of a System of Coagulating Brownian Particles
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…
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Phase Transition for the Speed of the Biased Random Walk on the Supercritical Percolation Cluster
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,…
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A PATCHWORK QUILT SEWN FROM BROWNIAN FABRIC: REGULARITY OF POLYMER WEIGHT PROFILES IN BROWNIAN LAST PASSAGE PERCOLATION
  • A. Hammond
  • Mathematics
    Forum 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…
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Biased random walks on a Galton-Watson tree with leaves
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…
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Fractal geometry of Airy$_{2}$ processes coupled via the Airy sheet
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…
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