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Finitely dependent coloring of \Z
Thursday, 9 November, 2023 - 17:00
Résumé :
A q-coloring of \Z is a way (=a probability measure) to assign a color to each integer, among a choice of q colors, with the condition that no two consecutive integers have the same color.
In other words, we are looking at sequences (X_i)_{i \in \Z} of {1, ..., q}-valued random variable wich satisfies X_i \neq X_{i+1} almost surely.
Since we only ask for local constraints, we may think that if i and j are two far enough integers, X_i and X_j should be independent ; we call the process k-dependent if this property holds as soon as |i-j|>k.
The main question of this talk will be : for which k and q does there exist a (stationary) k-dependent q-coloring of the integers ?
Beyond the trivial cases, Schramm proved that there is no 1-dependent 3-coloring and conjectured that the same holds for any k and q.
However, Holroyd and Liggett constructed a 1-dependent 4-coloring and a 2-dependent 3-coloring of the integers, disproving Schramm's conjecture.
Institution de l'orateur :
Institut Fourier
Thème de recherche :
Compréhensible
Salle :
4
