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Combinatoire de l’ASEP, arbres non-ambigus et polyominos parallélogrammes périodiques

Abstract : This thesis deals with a combinatorial interpretation of the stationnarydistribution of the ASEP given by staircase tableaux and studiestwo combinatorial objects : non-ambiguous trees and periodic parallelogrampolyominoes.In the first part, we study the matrix ansatz introduced by Derrida, Evans,Hakim and Pasquier. Any solution of this equation system can be used tocompute the stationnary probabilities of the ASEP. Our work defines newrecurrences equivalent to the matrix ansatz. By defining an insertion algorithmfor staircase tableaux, we prove combinatorially and easily that they satisfyour new recurrences. We do the same for the ASEP with two types of particles.Finally, we enumerate the corners of the tableaux related to the ASEP, whichgives the average number of transitions from a state of the ASEP.In the second part, we compute nice formulas for the generating functionsof non-ambiguous trees, from which we deduce enumeration formulas. Then, wegive a combinatorial interpretation of some of our results. Lastly, we generalisenon-ambiguous trees to every finite dimension.In the last part, we define a tree structure in periodic parallelogram polyominoes,motivated by the work of Boussicault, Rinaldi and Socci. It allowsus to compute easily the generating function with respect to the height andthe width as well as two new statistics : the intrinsic width and the intrinsicgluing height. Finally, we investigate the ultimate periodicity of the generatingfunction with respect to the area.
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Patxi Laborde-Zubieta. Combinatoire de l’ASEP, arbres non-ambigus et polyominos parallélogrammes périodiques. Autre [cs.OH]. Université de Bordeaux, 2017. Français. ⟨NNT : 2017BORD0709⟩. ⟨tel-01679732⟩

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