The cutoff profile for the simple exclusion process on the circle

• Hubert Lacoin ^[1]
  1. [1] Instituto Nacional de Matemática Pura e Aplicada

     Instituto Nacional de Matemática Pura e Aplicada

     Brasil

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 5, 2016, págs. 3399-3430
• Idioma: inglés
• DOI: 10.1214/15-AOP1053
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• Resumen

  • In this paper, we give a very accurate description of the way the simple exclusion process relaxes to equilibrium. Let PtPt denote the semi-group associated the exclusion on the circle with 2N2N sites and NN particles. For any initial condition χχ, and for any t≥4N29π2logNt≥4N29π2log⁡N, we show that the probability density Pt(χ,⋅)Pt(χ,⋅) is given by an exponential tilt of the equilibrium measure by the main eigenfunction of the particle system. As 4N29π2logN4N29π2log⁡N is smaller than the mixing time which is N22π2logNN22π2log⁡N, this allows to give a sharp description of the cutoff profile: if dN(t)dN(t) denote the total-variation distance starting from the worse initial condition we have limN→∞dN(N22π2logN+N2π2s)=erf(2–√πe−s), limN→∞dN(N22π2log⁡N+N2π2s)=erf⁡(2πe−s), where erferf is the Gauss error function.