Cube Root Asymptotics

Second class particles and cube root asymptotics for Hammersley's process

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Second Class Particles and Cube Root Asymptotics for Hammersley’s Process by Eric Cator

We show that, for a stationary version of Hammersley’s process, with Poisson sources on the positive x-axis and Poisson sinks on the positive y-axis, the variance of the length of a longest weakly North–East path L(t, t) from (0,0) to (t, t) is equal to 2E(t −X(t))+, where X(t) is the location of a second class particle at time t . This implies that both E(t −X(t))+ and the variance of L(t, t) ...

Cube{root Boundary Fluctuations for Droplets in Random Cluster Models

For a family of bond percolation models on Z that includes the FortuinKasteleyn random cluster model, we consider properties of the “droplet” that results, in the percolating regime, from conditioning on the existence of an open dual circuit surrounding the origin and enclosing at least (or exactly) a given large area A. This droplet is a close surrogate for the one obtained by Dobrushin, Kotec...

Planar graphs with square or cube root are four-colorable

Solving the LPN problem in cube-root time

In this paper it is shown that given a sufficient number of (noisy) random binary linear equations, the Learning from Parity with Noise (LPN) problem can be solved in essentially cube root time in the number of unknowns. The techniques used to recover the solution are known from fast correlation attacks on stream ciphers. As in fast correlation attacks, the performance of the algorithm depends ...