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End date: 2019 × Start date: 2010 × DDC: 510 × Publisher: [Madralin] : EMIS ELibEMS × WGL Institute: WIAS ×

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    Localization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model
    ([Madralin] : EMIS ELibEMS, 2018) Flegel, Franziska
    We study the asymptotic behavior of the principal eigenvector and eigenvalue of the random conductance Laplacian in a large domain of Zd (d≥2) with zero Dirichlet condition. We assume that the conductances w are positive i.i.d. random variables, which fulfill certain regularity assumptions near zero. If γ=sup{q≥0:E[w−q]<∞}<1/4, then we show that for almost every environment the principal Dirichlet eigenvector asymptotically concentrates in a single site and the corresponding eigenvalue scales subdiffusively. The threshold γc=1/4 is sharp. Indeed, other recent results imply that for γ>1/4 the top of the Dirichlet spectrum homogenizes. Our proofs are based on a spatial extreme value analysis of the local speed measure, Borel-Cantelli arguments, the Rayleigh-Ritz formula, results from percolation theory, and path arguments.
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    Scaling limit of ballistic self-avoiding walk interacting with spatial random permutations
    ([Madralin] : EMIS ELibEMS, 2019) Betz, Volker; Taggi, Lorenzo
    We consider nearest neighbour spatial random permutations on Zd. In this case, the energy of the system is proportional to the sum of all cycle lengths, and the system can be interpreted as an ensemble of edge-weighted, mutually self-avoiding loops. The constant of proportionality, α, is the order parameter of the model. Our first result is that in a parameter regime of edge weights where it is known that a single self-avoiding loop is weakly space filling, long cycles of spatial random permutations are still exponentially unlikely. For our second result, we embed a self-avoiding walk into a background of spatial random permutations, and condition it to cover a macroscopic distance. For large values of α (where long cycles are very unlikely) we show that this walk collapses to a straight line in the scaling limit, and give bounds on the fluctuations that are almost sufficient for diffusive scaling. For proving our results, we develop the concepts of spatial strong Markov property and iterative sampling for spatial random permutations, which may be of independent interest. Among other things, we use them to show exponential decay of correlations for large values of α in great generality.
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    Extremal decomposition for random Gibbs measures: from general metastates to metastates on extremal random Gibbs measures
    ([Madralin] : EMIS ELibEMS, 2018) Cotar, Codina; Jahnel, Benedikt; Külske, Christof
    The concept of metastate measures on the states of a random spin system was introduced to be able to treat the large-volume asymptotics for complex quenched random systems, like spin glasses, which may exhibit chaotic volume dependence in the strong-coupling regime. We consider the general issue of the extremal decomposition for Gibbsian specifications which depend measurably on a parameter that may describe a whole random environment in the infinite volume. Given a random Gibbs measure, as a measurable map from the environment space, we prove measurability of its decomposition measure on pure states at fixed environment, with respect to the environment. As a general corollary we obtain that, for any metastate, there is an associated decomposition metastate, which is supported on the extremes for almost all environments, and which has the same barycenter.