Anita Kollwitz

Matthew Dickson, LMU München: The Triangle Condition for the Marked Random Connection Model (Oberseminar Mathematische Stochastik)

Wednesday, 27.04.2022 17:00 im Raum M4

Mathematik und Informatik

The marked random connection model (RCM) is a random graph model that includes both geometric behaviour from giving vertices a spatial coordinate (in d-dimensional Euclidean space), and heterogeneity from giving the vertices a 'mark' coordinate in some probability space. For the standard (non-marked) RCM, the "Triangle Condition" is a powerful tool that has proven many results to hold at and around the critical transition in sufficiently high dimensions. These have included the proof that certain critical exponents take their so-called "mean-field" values, and that percolation does not occur at criticality. Here I present a generalisation of a previous result from (Heydenreich, van der Hofstad, Last, Matzke '20) for the non-marked model, and prove that the triangle condition is satisfied by a wide variety of marked RCMs. Both this previous result and the new result also prove an infra-red bound for the two-point function (hence describing long-range connectivity behaviour). I will outline the proof for these results which utilise a lace expansion argument. The main innovation here is a formulation in terms of self-adjoint linear operators which enables us to incorporate the extra mark behaviour. Joint work with Markus Heydenreich.

Angelegt am Wednesday, 30.03.2022 14:20 von Anita Kollwitz
Geändert am Wednesday, 20.04.2022 09:56 von Anita Kollwitz
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