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In a recent paper by the same authors, we constructed a stationary 1-dependent 4-coloring of the integers that is invariant under permutations of the colors. This was the first stationary k-dependent q-coloring for any k and q. When the analogous construction is carried out for q>4 colors, the resulting process is not k-dependent for any k. We construct here a process that is symmetric in the colors and 1-dependent for every q>=4. The construction uses a recursion involving Chebyshev polynomials evaluated at $sqrt{q}/2$.
This paper is concerned with symmetric $1$-dependent colorings of the $d$-ray star graph $mathscr{S}^d$ for each $d ge 2$. We compute the critical point of the $1$-dependent hard-core processes on $mathscr{S}^d$, which gives a lower bound for the num
Polypolyhedra (after R. Lang) are compounds of edge-transitive 1-skeleta. There are 54 topologically different polypolyhedra, and each has icosidodecahedral, cuboctahedral, or tetrahedral symmetry, all are realizable as modular origami models with on
We prove that proper coloring distinguishes between block-factors and finitely dependent stationary processes. A stochastic process is finitely dependent if variables at sufficiently well-separated locations are independent; it is a block-factor if i
We prove, under an assumption on the critical points of a real-valued function, that the symmetric Ising perceptron exhibits the `frozen 1-RSB structure conjectured by Krauth and Mezard in the physics literature; that is, typical solutions of the mod
We compute the best constant in the Khintchine inequality under assumption that the sum of Rademacher random variables is zero.