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Lengths of monotone subsequences in a Mallows permutation
We study the length of the longest increasing and longest decreasing subsequences of random permutations drawn from the Mallows measure. Under this measure, the probability of a permutation $$\uppi…
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On the Cycle Structure of Mallows Permutations
We study the length of cycles of random permutations drawn from the Mallows distribution. Under this distribution, the probability of a permutation $\pi \in \mathbb{S}_n$ is proportional to…
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Gravitational allocation to Poisson points
For d ≥ 3, we construct a non-randomized, fair and translationequivariant allocation of Lebesgue measure to the points of a standard Poisson point process in R d , defined by allocating to each of…
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High-Dimensional Lipschitz Functions are Typically Flat
A homomorphism height function on the d-dimensional torus Z_n^d is a function on the vertices of the torus taking integer values and constrained to have adjacent vertices take adjacent integer…
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On the transition between the disordered and antiferroelectric phases of the 6-vertex model.
The symmetric six-vertex model with parameters $a,b,c>0$ is expected to exhibit different behavior in the regimes $a+b c$ (ferroelectric). In this work, we study the way in which the transition…
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Longest increasing path within the critical strip
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…
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Probabilistic existence of regular combinatorial structures
TLDR
This work shows the existence of regular combinatorial objects which previously were not known to exist for a wide range of the underlying parameters, and proves that a randomly chosen structure has the required properties with positive yet tiny probability.
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Exponential Decay of Correlations in the 2D Random Field Ising Model
An extension of the Ising spin configurations to continuous functions is used for an exact representation of the random field Ising model’s order parameter in terms of disagreement percolation. This…
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Phase Transitions in Gravitational Allocation
Given a Poisson point process of unit masses (“stars”) in dimension d ≥ 3, Newtonian gravity partitions space into domains of attraction (cells) of equal volume. In earlier work, we showed the…
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Probabilistic existence of rigid combinatorial structures
TLDR
This work shows the existence of rigid combinatorial objects which previously were not known to exist for a wide range of the underlying parameters, including non-trivial orthogonal arrays, t-designs, and t-wise permutations.
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