The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity
We prove that the Tutte embeddings (aka harmonic/barycentric embeddings) of certain
random planar maps converge to γ-Liouville quantum gravity (γ-LQG). Specifically, we treat …
random planar maps converge to γ-Liouville quantum gravity (γ-LQG). Specifically, we treat …
Anomalous diffusion of random walk on random planar maps
E Gwynne, T Hutchcroft - Probability Theory and Related Fields, 2020 - Springer
We prove that the simple random walk on the uniform infinite planar triangulation (UIPT)
typically travels graph distance at most n^ 1/4+ o_n (1) n 1/4+ on (1) in n units of time …
typically travels graph distance at most n^ 1/4+ o_n (1) n 1/4+ on (1) in n units of time …
Supercritical percolation on nonamenable graphs: Isoperimetry, analyticity, and exponential decay of the cluster size distribution
J Hermon, T Hutchcroft - Inventiones mathematicae, 2021 - Springer
Let G be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation
on G. We prove that if G is nonamenable and p> p_c (G) p> pc (G) then there exists a …
on G. We prove that if G is nonamenable and p> p_c (G) p> pc (G) then there exists a …
Peeling random planar maps
N Curien - Saint-Flour lecture notes, 2019 - Springer
When thinking of planar lattices, the reader may immediately come up with “Euclidean”
planar graphs such as. Z2, the triangular grid, or a hyperbolic lattice. Those examples are …
planar graphs such as. Z2, the triangular grid, or a hyperbolic lattice. Those examples are …
Local limits of bipartite maps with prescribed face degrees in high genus
T Budzinski, B Louf - The Annals of Probability, 2022 - projecteuclid.org
We study the local limits of uniform high genus bipartite maps with prescribed face degrees.
We prove the convergence toward a family of infinite maps of the plane, the q-IBPMs, which …
We prove the convergence toward a family of infinite maps of the plane, the q-IBPMs, which …
Percolation on hyperbolic graphs
T Hutchcroft - Geometric and Functional Analysis, 2019 - Springer
We prove that Bernoulli bond percolation on any nonamenable, Gromov hyperbolic, quasi-
transitive graph has a phase in which there are infinitely many infinite clusters, verifying a …
transitive graph has a phase in which there are infinitely many infinite clusters, verifying a …
Uniqueness of the infinite tree in low-dimensional random forests
N Halberstam, T Hutchcroft - arXiv preprint arXiv:2302.12224, 2023 - arxiv.org
The arboreal gas is the random (unrooted) spanning forest of a graph in which each forest is
sampled with probability proportional to $\beta^{\#\text {edges}} $ for some $\beta\geq 0 …
sampled with probability proportional to $\beta^{\#\text {edges}} $ for some $\beta\geq 0 …
Uniform even subgraphs and graphical representations of Ising as factors of iid
We prove that the Loop O (1) model, a well-known graphical expansion of the Ising model, is
a factor of iid on unimodular random rooted graphs under various conditions, including in …
a factor of iid on unimodular random rooted graphs under various conditions, including in …
The number of ends in the uniform spanning tree for recurrent unimodular random graphs
D van Engelenburg, T Hutchcroft - arXiv preprint arXiv:2301.03875, 2023 - arxiv.org
We prove that if a unimodular random rooted graph is recurrent, the number of ends of its
uniform spanning tree is almost surely equal to the number of ends of the graph. Together …
uniform spanning tree is almost surely equal to the number of ends of the graph. Together …
Unimodular hyperbolic triangulations: circle packing and random walk
We show that the circle packing type of a unimodular random plane triangulation is
parabolic if and only if the expected degree of the root is six, if and only if the triangulation is …
parabolic if and only if the expected degree of the root is six, if and only if the triangulation is …