2011/10/26 by Quentin Berger, Berger, Quentin, Fabio Toninelli +1
Mathematics · Physics and Astronomy · #60K37 #82B44 #82D60 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.MP #math.PR #msc:60K37 #msc:82B44 #msc:82D60
paper · pdf · doi:10.48550/arxiv.1110.5781
38 pages, 3 figures
arxiv created 2011/10/26 · arxiv updated 2011/10/27
We consider the hierarchical disordered pinning model studied in [9], which exhibits a localization/delocalization phase transition. In the case where the disorder is i.i.d. (independent and identically distributed), the question of relevance/irrelevance of disorder (i.e. whether disorder changes or not the critical properties with respect to the homogeneous case) is by now mathematically rather well understood [14,15]. Here we consider the case where randomness is spatially correlated and correlations respect the hierarchical structure of the model; in the non-hierarchical model our choice would correspond to a power-law decay of correlations. In terms of the critical exponent of the homogeneous model and of the correlation decay exponent, we identify three regions. In the first one (non-summable correlations) the phase transition disappears. In the second one (correlations decaying fast enough) the system behaves essentially like in the i.i.d. setting and the relevance/irrelevance criterion is not modified. Finally, there is a region where the presence of correlations changes the critical properties of the annealed system.