We study longest and shortest directed paths in the following natural model of directed random planar maps: the uniform infinite bipolar-oriented triangulation (UIBOT), which is the local limit of uniform bipolar-oriented triangulations around a typical edge. We construct the Busemann function which measures directed distance to along a natural interface in the UIBOT. We show …
We investigate the asymptotic properties of permutations drawn from the Luce model, a natural probabilistic framework in which permutations are generated sequentially by sampling without replacement, with selection probabilities proportional to prescribed positive weights. These permutations arise in applications such as ranking models, the Tsetlin library, and related Markov processes. Under minimal assumptions on the …
In the context of two-dimensional large- lattice Yang–Mills theory, we perform a refined study of the surface sums defined in the companion work [19]. In this setting, the surface sums are a priori expected to exhibit significant simplifications, because two-dimensional Yang–Mills theory is a special model admitting many known exact formulas. Thus, a natural problem is …
We establish a scaling limit result for the length LIS of the longest increasing subsequence of a permutation of size sampled from the Brownian separable permuton of parameter , which is the universal limit of pattern-avoiding permutations. Specifically, we prove that where is the unique solution in the interval to the equation and is a non-deterministic …
Permutons, which are probability measures on the unit square with uniform marginals, are the natural scaling limits for sequences of (random) permutations. We introduce a -dimensional generalization of these measures for all , which we call –dimensional permutons, and extend — from the two-dimensional setting — the theory to prove convergence of sequences of (random) …
[20] High-dimensional permutons: theory and applications (with Andrew Lin).Read More »
We give a sum over weighted planar surfaces formula for Wilson loop expectations in the large- limit of strongly coupled lattice Yang–Mills theory, in any dimension. The weights of each surface are simple and expressed in terms of products of signed Catalan numbers. In establishing our results, the main novelty is to convert a recursive …
Permutons constructed from a Liouville quantum gravity surface and a pair of space-filling Schramm-Loewner evolutions (SLEs) have been shown — or are conjectured — to describe the scaling limit of various natural models of random constrained permutations. We prove that, in two distinct and natural settings, these permutons uniquely determine, modulo rotation, scaling, translation and …