I am a Herchel Smith Postdoctoral Research Fellow in the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge and a Junior Research Fellow in Trinity College. Previously, I obtained my PhD in the probability group in the University of British Columbia in Vancouver under the supervision of Asaf Nachmias and Omer Angel. Before that, I completed my BA and MMath at King's College, Cambridge.

I was an intern in the Microsoft Research Theory Group in the summers of 2015, 2016, and 2017, and was supported during my PhD by a Microsoft Research PhD Fellowship.

From July 2021 I will be a Professor of Mathematics at Caltech.

- Course notes are available here.
- Example sheet 1. Solutions.
- Example sheet 2. Solutions.
- Example sheet 3.

Probability: percolation, spin systems, self-avoiding walk, probability on groups, spanning forests, sandpiles, planar graphs, circle packing, random maps, random walk, potential theory, finitely dependent processes, random permutations.

- Uniform spanning forests in high dimension, 3 lecture mini-course, OOPS 2020. Notes.
- An operator-theoretic approach to nonamenable percolation, INI 2018.
- Interlacements and the uniform spanning forest, CIRM 2017.
- Percolation on nonunimodular graphs, Tel Aviv 2017
- The Strange Geometry of High-Dimensional Spanning Forests, MSR 2016.
- Circle Packing and Random Triangulations, MSR 2015.
- Indistinguishability of Trees in Uniform Spanning Forests, Banff 2015.
- Reversible Hyperbolic Triangulations: Circle Packing and Random Walk, Banff 2014.

- Gabor Pete:
*Kazhdan groups have cost 1*, Tel Aviv 2018. - Nicolas Curien:
*Geometric and spectral properties of causal maps*, Tel Aviv 2017 - Yuval Peres :
*The strange geometry of high-dimensional spanning forests*, MIT 2017. - Omer Angel:
*Parabolic and Hyperbolic Unimodular Maps*, INI 2015. - Asaf Nachmias:
*Uniform Spanning Forests of Planar Graphs*, INI 2015.

*Non-triviality of the phase transition for percolation on finite transitive graphs*, with Matthew Tointon. Preprint.

[ arXiv | pdf | abstract | BibTeX ]*The critical two-point function for long-range percolation on the hierarchical lattice*. Preprint.

[ arXiv | pdf | abstract | BibTeX ]*Transience and recurrence of sets for branching random walk via non-standard stochastic orders*.**Annales de l’Institut Henri Poincaré (AIHP)**, to appear.

[ arXiv | pdf | abstract | BibTeX ]*Logarithmic corrections to scaling in the four-dimensional uniform spanning tree*, with Perla Sousi. Preprint.

[ arXiv | pdf | abstract | BibTeX ]*Collisions of random walks in dynamic random environments*, with Noah Halberstam. Preprint.

[ arXiv | pdf | abstract | BibTeX ]*Power-law bounds for critical long-range percolation below the upper-critical dimension*.**Probability Theory and Related Fields**, special issue in memory of Harry Kesten, to appear.

[ arXiv | pdf | abstract | BibTeX ]*Continuity of the Ising phase transition on nonamenable groups*. Preprint.

[ arXiv | pdf | abstract | BibTeX ]*On the tail of the branching random walk local time*, with Omer Angel and Antal Jarai.**Probability Theory and Related Fields (PTRF)**, to appear.

[ arXiv | pdf | abstract | BibTeX ]*Slightly supercritical percolation on nonamenable graphs I: The distribution of finite clusters*. Preprint.

[ arXiv | pdf | abstract | BibTeX ]*Non-intersection of transient branching random walks*.**Probability Theory and Related Fields (PTRF)**, 178, 1–23(2020).

[ arXiv | pdf | abstract | BibTeX ]*Large, lengthy graphs look locally like lines*, with Itai Benjamini.**Bulletin of the London Mathematical Society**, to appear.

[ arXiv | pdf | abstract | BibTeX ]*Supercritical percolation on nonamenable graphs: Isoperimetry, analyticity, and exponential decay of the cluster size distribution*, with Jonathan Hermon.**Inventiones Mathematicae**, 2020.

[ arXiv | pdf | abstract | BibTeX ]*The L2 boundedness condition in nonamenable percolation*.**Electronic Journal of Probability**, Volume 25, (2020).

[ arXiv | pdf | abstract | BibTeX ]*New critical exponent inequalities for percolation and the random cluster model*.**Probability and Mathematical Physics (PMP)**, to appear.

[ arXiv | pdf | abstract | BibTeX ]*Kazhdan groups have cost 1*, with Gabor Pete.**Inventiones Mathematicae**, 221, 873–891, (2020).

[ arXiv | pdf | abstract | BibTeX ]*No percolation at criticality on certain groups of intermediate growth*, with Jonathan Hermon.**International Mathematics Research Notices (IMRN)**, December 2019. 1073-7928.

[ arXiv | pdf | abstract | BibTeX ]*Locality of the critical probability for transitive graphs of exponential growth*.**Annals of Probability**, Volume 48, Number 3 (2020), 1352-1371.

[ arXiv | pdf | abstract | BibTeX ]*Anomalous diffusion of random walks on random planar maps*, with Ewain Gwynne.**Probability Theory and Related Fields (PTRF)**, 2020.

[ arXiv | pdf | abstract | BibTeX ]*Percolation on hyperbolic graphs*.**Geometric and Functional Analysis (GAFA)**, 29(3):766–810, 2019.

[ arXiv | pdf | abstract | BibTeX ]*Universality of high-dimensional spanning forests and sandpiles*.**Probability Theory and Related Fields (PTRF)**, 176(1-2):533–597, 2020.

[ arXiv | pdf | abstract | BibTeX ]*Coalescing random walk on unimodular graphs*, with Eric Foxall and Matthew Junge.**Electronic Communications in Probability (ECP)**, Volume 23 (2018), paper no. 62, 10 pp.

[ arXiv | pdf | abstract | BibTex ]*Mallows permutations as stable matchings*, with Omer Angel, Alexander E. Holroyd and Avi Levy.**Canadian Journal of Mathematics**, to appear.

[ arXiv | pdf | abstract | BibTeX ]*Statistical physics on a product of trees*.**Annales de l’Institut Henri Poincaré (AIHP)**, Volume 55, Number 2 (2019), 1001-1010.

[ arXiv | pdf | abstract | BibTeX ]*Non-uniqueness and mean-field criticality for percolation on nonunimodular transitive graphs*.**Journal of the American Mathematical Society (JAMS)**, 33 (2020), 1101-1165.

[ arXiv | pdf | abstract | BibTeX ]*Geometric and spectral properties of causal maps*, with Nicolas Curien and Asaf Nachmias.**Journal of the European Mathematical Society (JEMS)**, to appear.

[ arXiv | pdf | abstract | BibTeX ]*Counterexamples for percolation on unimodular random graphs*, with Omer Angel.**Unimodularity in randomly generated graphs**, 11–28, Contemp. Math., 719, Amer. Math. Soc., Providence, RI, 2018.

[ arXiv | pdf | abstract | BibTeX ]*Self-avoiding walk on nonunimodular transitive graphs*.**Annals of Probability**, Volume 47, Number 5 (2019), 2801-2829.

[ arXiv | pdf | abstract | BibTeX ]*The Hammersley-Welsh bound for self-avoiding walk revisited*.**Electronic Communications in Probability (ECP)**, Volume 23 (2018), paper no. 5, 8 pp.

[ arXiv | pdf | abstract | BibTeX ]*Finitely Dependent Cycle Coloring*, with Alexander E. Holroyd and Avi Levy.**Electronic Communications in Probability (ECP)**, Volume 23 (2018), paper no. 64, 12 pp.

[ arXiv | pdf | abstract | BibTeX ]*Harmonic Dirichlet Functions on Planar Graphs*.**Discrete and Computational Geometry**, 61(3):479–506, 2019.

[ arXiv | pdf | abstract | BibTeX ]*Mallows Permutations and Finite Dependence*, with Alexander E. Holroyd and Avi Levy.**Annals of Probability**, Volume 48, Number 1 (2020), 343-379.

[ arXiv | pdf | abstract | BibTeX ]*Indistinguishability of collections of trees in the uniform spanning forest*.**Annales de l’Institut Henri Poincaré (AIHP)**, Volume 56, Number 2 (2020), 917-927.

[ arXiv | pdf | abstract | BibTeX ]*The Component Graph of the Uniform Spanning Forest: Transitions in Dimensions 9, 10, 11, ...*

with Yuval Peres.**Probability Theory and Related Fields (PTRF)**, 175(1-2):141–208, 2019.

[ arXiv | pdf | abstract | BibTeX ]-
*Hyperbolic and Parabolic Unimodular Random Maps*, with Omer Angel, Asaf Nachmias

and Gourab Ray.**Geometric and Functional Analysis (GAFA)**, July 2018, Volume 28, Issue 4, pp 879–942.

[ arXiv | pdf | abstract | BibTeX ] -
*Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters.***Comptes Rendus Mathematique**, Volume 354, Issue 9, Pages 944-947.

[ arXiv | pdf | abstract | BibTeX ] -
*Uniform Spanning Forests of Planar Graphs.*with Asaf Nachmias.**Forum of Mathematics, Sigma**, 7:e29, 55, 2019.

[ arXiv | pdf | abstract | BibTeX ] -
*Interlacements and the Wired Uniform Spanning Forest.***Annals of Probability**, Volume 46, Number 2 (2018), 1170-1200.

[ arXiv | pdf | abstract | BibTeX ] -
*Boundaries of Planar Graphs: A Unified Approach.*with Yuval Peres.**Electronic Journal of Probability (EJP)**, Volume 22 (2017), paper no. 100, 20 pp.

[ arXiv | pdf | abstract | BibTeX ] -
*Indistinguishability of Trees in Uniform Spanning Forests.*with Asaf Nachmias.**Probability Theory and Related Fields (PTRF)**, June 2017, Volume 168, Issue 1–2, pp 113–152.

[ arXiv | pdf | abstract | BibTeX ] -
*Collisions of Random Walks in Reversible Random Graphs.*with Yuval Peres.**Electronic Communications in Probability (ECP)**, 20, no. 63, 1-6, 2015.

[ arXiv | pdf | abstract | BibTeX ] -
*Wired Cycle-Breaking Dynamics for Uniform Spanning Forests.***Annals of Probability**, 44 (2016), no. 6, 3879-3892.

[ arXiv | pdf | abstract | BibTeX ] -
*Unimodular Hyperbolic Triangulations: Circle Packing and Random Walk*, with Omer Angel,

Asaf Nachmias and Gourab Ray.**Inventiones Mathematicae**, (2016) 206: 229. doi:10.1007/s00222-016-0653-9.

[ arXiv | pdf | abstract | BibTeX ]

- My PhD thesis,
*Discrete Probability and the Geometry of Graphs*, is available here and through the UBC library.