Random Sorting Networks, the Directed Landscape, and Random Polynomials

The first part of this thesis concerns random sorting networks. A sorting network is a shortest path from 12···n to n···21 in the Cayley graph of the symmetric group Sn generated by adjacent transpositions. We prove that in a uniform random n-element sorting network σn, all particle trajectories are close to sine curves with high probability. We also find the weak limit of the time-t permutation matrix measures of sign. As a corollary, we show that if Sn is embedded into Rn via the map τ→(τ(1),τ(2),...τ(n)), then with high probability, σn is close to a great circle on a particular (n-2)-dimensional sphere. These results prove conjectures of Angel, Holroyd, Romik, and Vir'ag. To prove these results, we first find a local limit and prove that local speeds follow an arcsine distribution. The second part of this thesis concerns last passage percolation. The conjectured limit of last passage percolation is a scale-invariant, independent, stationary-increment process with respect to metric composition. We prove this for Brownian last passage percolation. We construct the Airy sheet and characterize it in terms of the Airy line ensemble. We also show that last passage geodesics converge to random Hölder-2/3- continuous functions. To prove these results, we develop a new probabilistic framework for understanding the Airy line ensemble. The third part of this thesis concerns random polynomials. Let Gn = ∑ni=0 ξipi, where the ξi are i.i.d. non-degenerate complex random variables, and {pi} is a sequence of orthonormal polynomials with respect to a regular measure τ supported on a compact set K. We show that the zero measure of Gn converges weakly almost surely to the equilibrium measure of K if and only if Elog(1 + |ξ0|) < ∞, and that the zero measure of Gn converges weakly in probability to the equilibrium measure of K if and only if P(|ξ0| > en) = o(n-1). Our methods also work for more general sequences of asymptotically minimal polynomials in Lp(τ), where p ∈ (0, ∞].

Department dc:contributor.department

Mathematics

Year dc:date.issued

2019

Author dc:creator

• Dauvergne, Duncan

Advisor dc:contributor.advisor

• Virag, Balint

• directed landscape
• KPZ universality class
• last passage percolation
• random polynomials
• random sorting networks
• Young tableau

Handle dc:identifier.uri

http://hdl.handle.net/1807/95778

OAI identifier oai:identifier

oai:utoronto.scholaris.ca:1807/95778