Tom Hutchcroft

Email: t • hutchcroft ⓐ maths • cam • ac • uk

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.

I plan to go on the job market in the second half of 2020. Click here for my CV.

Organization

Teaching

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

Research interests

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.

Online Talks

• 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.

Talks by co-authors

• 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.

Papers

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

  We study dynamic random conductance models on Z^2 in which the environment evolves as a reversible Markov process that is stationary under space-time shifts. We prove under a second moment assumption that two conditionally independent random walks in the same environment collide infinitely often almost surely. These results apply in particular to random walk on dynamical percolation.

  @article{HalberstamHutchcroft2020,
    title={Collisions of random walks in dynamic random environments},
    author={Halberstam, Noah and Hutchcroft, Tom},
    journal={arXiv preprint arXiv:2009.13951},
    year={2020}
  }
  

• Power-law bounds for critical long-range percolation below the upper-critical dimension. Preprint.
  [ arXiv | pdf | abstract | BibTeX ]

  We study long-range Bernoulli percolation on Zd in which each two vertices x and y are connected by an edge with probability 1−exp(−β∥x−y∥^(−d−α)). It is a theorem of Noam Berger (CMP, 2002) that if 0 < α < d then there is no infinite cluster at the critical parameter βc. We give a new, quantitative proof of this theorem establishing the power-law upper bound P_{βc}(|K|≥n)≤Cn^(−(d−α)/(2d+α)) for every n≥1, where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour.

  As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality (2−η)(δ+1)≤d(δ−1) relating the cluster-volume exponent δ and two-point function exponent η.

  @article{hutchcroft2020power,
    title={Power-law bounds for critical long-range percolation below the upper-critical dimension},
    author={Hutchcroft, Tom},
    journal={arXiv preprint arXiv:2008.11197},
    year={2020}
  }
  

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

  We prove rigorously that the ferromagnetic Ising model on any nonamenable Cayley graph undergoes a continuous (second-order) phase transition in the sense that there is a unique Gibbs measure at the critical temperature. The proof of this theorem is quantitative and also yields power-law bounds on the magnetization at and near criticality. Indeed, we prove more generally that the magnetization is a locally Hölder-continuous function of the inverse temperature and external field throughout the non-negative quadrant. As a second application of the methods we develop, we also prove that the free energy of Bernoulli percolation is twice differentiable at pc on any transitive nonamenable graph.

  @article{hutchcroft2020continuity,
    title={Continuity of the Ising phase transition on nonamenable groups},
    author={Hutchcroft, Tom},
    journal={arXiv preprint arXiv:2007.15625},
    year={2020}
  }
  

• 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 ]

  Consider a critical branching random walk on Zd, d≥1, started with a single particle at the origin, and let L(x) be the total number of particles that ever visit a vertex x. We study the tail of L(x) under suitable conditions on the offspring distribution. In particular, our results hold if the offspring distribution has an exponential moment.

  @article{2002.12188
  Author = {Omer Angel and Tom Hutchcroft and Antal A. Járai},
  Title = {On the tail of the branching random walk local time},
  Year = {2020},
  Note = {Preprint. Available at \url{https://arxiv.org/abs/2002.12188}},
  }
  

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

  We study the distribution of finite clusters in slightly supercritical (p↓pc) Bernoulli bond percolation on transitive nonamenable graphs, proving in particular that if G is a transitive nonamenable graph satisfying the L2 boundedness condition and K denotes the cluster of the origin then there exists δ>0 such that Pp(n≤|K|<∞)≍n^(−1/2)exp[−Θ(|p−pc|2n)] and Pp(r≤Rad(K)<∞)≍r^(−1)exp[−Θ(|p−pc|r)] for every p∈(pc−δ,pc+δ) and n,r≥1, where all implicit constants depend only on G. We deduce in particular that the critical exponents γ′ and Δ′ describing the rate of growth of the moments of a finite cluster as p↓pc take their mean-field values of 1 and 2 respectively.

  These results apply in particular to Cayley graphs of nonelementary hyperbolic groups, to products with trees, and to transitive graphs of spectral radius ρ<1/2. In particular, every finitely generated nonamenable group has a Cayley graph to which these results apply. They are new for graphs that are not trees. The corresponding facts are yet to be understood on Zd even for d very large. In a second paper in this series, we will apply these results to study the geometric and spectral properties of infinite slightly supercritical clusters in the same setting.

  @article{2002.02916
  Author = {Tom Hutchcroft},
  Title = {Slightly supercritical percolation on nonamenable graphs I: The distribution of finite clusters},
  Year = {2020},
  Note = {Preprint. Available at \url{https://arxiv.org/abs/2002.02916}},
  }
  

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

  Let G be a Cayley graph of a nonamenable group with spectral radius ρ < 1. It is known that branching random walk on G with offspring distribution μ is transient, i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offsring is at most 1/ρ. Benjamini and Müller (2010) conjectured that throughout the transient supercritical phase 1<μ¯≤ρ−1, and in particular at the recurrence threshold μ¯=ρ−1, the trace of the branching random walk is tree-like in the sense that it is infinitely-ended almost surely on the event that the walk survives forever. This is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely. We prove this conjecture, along with several other related conjectures made by the same authors.

  @article{1910.01018
  Author = {Tom Hutchcroft},
  Title = {Non-intersection of transient branching random walks},
  journal={Probability Theory and Related Fields}
  Year = {2020},
  Note = {To appear. Available at \url{https://arxiv.org/abs/1910.01018}},
  }
  

• Large, lengthy graphs look locally like lines, with Itai Benjamini. Preprint.
  [ arXiv | pdf | abstract | BibTeX ]

  We apply the theory of unimodular random rooted graphs to study the metric geometry of large, finite, bounded degree graphs whose diameter is proportional to their volume. We prove that for a positive proportion of the vertices of such a graph, there exists a mesoscopic scale on which the graph looks like R in the sense that the rescaled ball is close to a line segment in the Gromov-Hausdorff metric

  @article{1905.00316,
  Author = {Itai Benjamini and Tom Hutchcroft},
  Title = {Large, lengthy graphs look locally like lines},
  Year = {2019},
  Note = {arXiv:1905.00316},
  }
  

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

  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) then there exists a positive constant cp such that P_p(n≤|K|<∞)≤e^(−cpn) for every n≥1, where K is the cluster of the origin. We deduce the following two corollaries:

  1. Every infinite cluster in supercritical percolation on a transitive nonamenable graph has anchored expansion almost surely. This answers positively a question of Benjamini, Lyons, and Schramm (1997).
  2. For transitive nonamenable graphs, various observables including the percolation probability, the truncated susceptibility, and the truncated two-point function are analytic functions of p throughout the entire supercritical phase.

  @article{1904.10448,
  Author = {Jonathan Hermon and Tom Hutchcroft},
  Title = {Supercritical percolation on nonamenable graphs: Isoperimetry, analyticity, and exponential decay of the cluster size distribution},
  Year = {2019},
  Note = {arXiv:1904.10448},
  }
  

• The L2 boundedness condition in nonamenable percolation. Electronic Journal of Probability, to appear.
  [ arXiv | pdf | abstract | BibTeX ]

  Let G=(V,E) be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on G. In recent work, we conjectured that if G is nonamenable then the matrix of critical connection probabilities Tpc(u,v)=ℙpc(u↔v) is bounded as an operator Tpc:L2(V)→L2(V) and proved that this conjecture holds for several classes of graphs. We also noted in that work that the conjecture implies two older conjectures, namely that percolation on transitive nonamenable graphs always has a nontrivial nonuniqueness phase, and that critical percolation on the same class of graphs has mean-field critical behaviour.

  In this paper we further investigate the consequences of the L2 boundedness conjecture. In particular, we prove that the following hold for all transitive graphs: i) The two-point function decays exponentially in the distance for all p ; ii) If pc< p2→2, then the critical exponent governing the extrinsic diameter of a critical cluster is 1; iii) Below p2→2, percolation is "ballistic" in the sense that the intrinsic distance between two points is exponentially unlikely to be much larger than their extrinsic distance; iv) If pc < p_{2→2}, then ‖Tpc‖q→q≍(q−1)−1 and pq→q−pc≍q−1 as q↓1. v) If pc < p2→2, then various 'multiple-arm' events have probabilities comparable to the upper bound given by the BK inequality. In particular, the probability that the origin is a trifurcation point is of order (p−pc)^3 as p↓pc. All of these results are new even in the Gromov hyperbolic case.

  Finally, we apply these results together with duality arguments to compute the critical exponents governing the geometry of intrinsic geodesics at the uniqueness threshold of percolation in the hyperbolic plane.

  @article{1904.05804,
  Author = {Tom Hutchcroft},
  Title = {The $L^2$ boundedness condition in nonamenable percolation},
  Year = {2019},
  Note = {arXiv:1901.10363},
  }
  

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

  We apply a variation on the methods of Duminil-Copin, Raoufi, and Tassion to establish a new differential inequality applying to both Bernoulli percolation and the Fortuin-Kasteleyn random cluster model. This differential inequality has a similar form to that derived for Bernoulli percolation by Menshikov but with the important difference that it describes the distribution of the volume of a cluster rather than of its radius. We apply this differential inequality to prove the following:

  1. The critical exponent inequalities γ≤δ−1 and Δ≤γ+1 hold for percolation and the random cluster model on any transitive graph. These inequalities are new even in the context of Bernoulli percolation on ℤ^d, and are saturated in mean-field for Bernoulli percolation and for the random cluster model with q∈[1,2).
  2. The volume of a cluster has an exponential tail in the entire subcritical phase of the random cluster model on any transitive graph. This proof also applies to infinite-range models, where the result is new even in the Euclidean setting.

  @article{1901.10363,
  Author = {Tom Hutchcroft},
  Title = {New critical exponent inequalities for percolation and the random cluster model},
  Year = {2019},
  Note = {arXiv:1901.10363},
  }
  

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

  We prove that every countably infinite group with Kazhdan's property (T) has cost 1, answering a well-known question of Gaboriau. It remains open if they have fixed price 1.

  @article{1810.11015,
  Author = {Tom Hutchcroft and Gabor Pete},
  Title = {Kazhdan groups have cost 1},
  Year = {2018},
  Note = {arXiv:1810.11015},
  }
  

• 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 ]

  We prove that critical percolation has no infinite clusters almost surely on any unimodular quasi-transitive graph satisfying a return probability upper bound of the form p_n(v,v)≤exp[−Ω(n^γ)] for some γ>1/2. The result is new in the case that the graph is of intermediate volume growth.

  @article{1809.11112,
      author = {Hermon, Jonathan and Hutchcroft, Tom},
      title = "{No Percolation at Criticality on Certain Groups of Intermediate Growth}",
      journal = {International Mathematics Research Notices},
      year = {2019},
      month = {12},
      issn = {1073-7928},
      doi = {10.1093/imrn/rnz265},
      url = {https://doi.org/10.1093/imrn/rnz265},
      note = {rnz265},
      eprint = {https://academic.oup.com/imrn/advance-article-pdf/doi/10.1093/imrn/rnz265/31234168/rnz265.pdf},
  }
  

• 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 ]

  Around 2008, Schramm conjectured that the critical probabilities for Bernoulli bond percolation satisfy the following continuity property: If (G_n)_{n≥1} is a sequence of transitive graphs converging locally to a transitive graph G and limsup p_c(G_n) < 1 then p_c(G_n)→p_c(G) as n→∞. We verify this conjecture under the additional hypothesis that there is a uniform exponential lower bound on the volume growth of the graphs in question. The result is new even in the case that the sequence of graphs is uniformly nonamenable.

  In the unimodular case, this result is obtained as a corollary to the following theorem of independent interest: For every g>1 and M<∞, there exist positive constants C=C(g,M) and δ=δ(g,M) such that if G is a transitive unimodular graph with degree at most M and growth gr(G) := inf_{r≥1}|B(o,r)|^{1/r} ≥ g, then P_{p_c}(|K_o|≥n)≤Cn^{−δ} for every n≥1, where K_o is the cluster of the root vertex o. The proof of this inequality makes use of new universal bounds on the probabilities of certain two-arm events, which hold for every unimodular transitive graph.

  @article{1808.08940,
  Author = {Tom Hutchcroft},
  Title = {Locality of the critical probability for transitive graphs of exponential growth},
  Journal = {Annals of Probability},
  Note = {To appear.} 
  Year = {2018},
  Eprint = {arXiv:1808.08940},
  }
  

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

  We prove that the simple random walk on the uniform infinite planar triangulation (UIPT) typically travels graph distance at most n^{1/4+o(1)} in n units of time. Together with the complementary lower bound proven by Gwynne and Miller (2017) this shows that that the typical graph distance displacement of the walk after n steps is n^{1/4+o(1)}, as conjectured by Benjamini and Curien (2013). More generally, we show that the simple random walks on a certain family of random planar maps in the γ-Liouville quantum gravity (LQG) universality class for γ∈(0,2)---including spanning tree-weighted maps, bipolar-oriented maps, and mated-CRT maps---typically travels graph distance n^{1/d_γ+o(1)} in n units of time, where dγ is the growth exponent for the volume of a metric ball on the map, which was shown to exist and depend only on γ by Ding and Gwynne (2018). Since d_γ>2, this shows that the simple random walk on each of these maps is subdiffusive.

  Our proofs are based on an embedding of the random planar maps under consideration into ℂ wherein graph distance balls can be compared to Euclidean balls modulo subpolynomial errors. This embedding arises from a coupling of the given random planar map with a mated-CRT map together with the relationship of the latter map to SLE-decorated LQG.

  @article{1807.01512,
    title={Anomalous diffusion of random walk on random planar maps},
    author={Gwynne, Ewain and Hutchcroft, Tom},
    Eprint={arXiv:1807.01512},
    year={2018}
  }
  

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

  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 well-known conjecture of Benjamini and Schramm (1996) under the additional assumption of hyperbolicity. In other words, we show that p_c < p_u for any such graph. Our proof also yields that the triangle condition ∇_{p_c} < ∞ holds at criticality on any such graph, which is known to imply that several critical exponents exist and take their mean-field values. This gives the first family of examples of one-ended groups all of whose Cayley graphs are proven to have mean-field critical exponents for percolation.

  @article{1804.10191,
      AUTHOR = {Hutchcroft, Tom},
       TITLE = {Percolation on hyperbolic graphs},
     JOURNAL = {Geom. Funct. Anal.},
    FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {29},
        YEAR = {2019},
      NUMBER = {3},
       PAGES = {766--810},
        ISSN = {1016-443X},
         DOI = {10.1007/s00039-019-00498-0},
         URL = {https://doi.org/10.1007/s00039-019-00498-0},
  }
  

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

  We prove that the wired uniform spanning forest exhibits mean-field behaviour on a very large class of graphs, including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph. Several of our results are new even in the case of ℤ^d, d≥5. In particular, we prove that every tree in the forest has spectral dimension 4/3 and walk dimension 3 almost surely, and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are 1 and 1/2 respectively. (The past of a vertex in the uniform spanning forest is the finite component that is disconnected from infinity when that vertex is deleted from the forest.) We obtain as a corollary that the critical exponent governing the extrinsic diameter of the past is 2 on any transitive graph of at least five dimensional polynomial growth, and is 1 on any bounded degree nonamenable graph. We deduce that the critical exponents describing the diameter and total number of topplings in an avalanche in the Abelian sandpile model are 2 and 1/2 respectively for any transitive graph with polynomial growth of dimension at least five, and are 1 and 1/2 respectively for any bounded degree nonamenable graph.

  In the case of ℤ^d, d≥5, some of our results regarding critical exponents recover earlier results of Bhupatiraju, Hanson, and Jarai (2017). In this case, we improve upon their results by showing that the tail probabilities in question are described by the appropriate power laws to within constant-order multiplicative errors, rather than the polylogarithmic-order multiplicative errors present in that work.

  @article{1804.04120,
      AUTHOR = {Hutchcroft, Tom},
       TITLE = {Universality of high-dimensional spanning forests and
                sandpiles},
     JOURNAL = {Probab. Theory Related Fields},
    FJOURNAL = {Probability Theory and Related Fields},
      VOLUME = {176},
        YEAR = {2020},
      NUMBER = {1-2},
       PAGES = {533--597},
        ISSN = {0178-8051},
         DOI = {10.1007/s00440-019-00923-3},
         URL = {https://doi.org/10.1007/s00440-019-00923-3},
  }
  

• 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 ]

  Coalescing random walk on a unimodular random rooted graph for which the root has finite expected degree visits each site infinitely often almost surely. A corollary is that an opinion in the voter model on such graphs has infinite expected lifetime. Additionally, we deduce an adaptation of our main theorem that holds uniformly for coalescing random walk on finite random unimodular graphs with degree distribution stochastically dominated by a probability measure with finite mean.

  @article{foxall2018,
  author = "Foxall, Eric and Hutchcroft, Tom and Junge, Matthew",
  doi = "10.1214/18-ECP136",
  fjournal = "Electronic Communications in Probability",
  journal = "Electron. Commun. Probab.",
  pages = "10 pp.",
  pno = "62",
  publisher = "The Institute of Mathematical Statistics and the Bernoulli Society",
  title = "Coalescing random walk on unimodular graphs",
  url = "https://doi.org/10.1214/18-ECP136",
  volume = "23",
  year = "2018"
  }
  

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

  We show that the Mallows measure on permutations of 1,…,n arises as the law of the unique Gale-Shapley stable matching of the random bipartite graph conditioned to be perfect, where preferences arise from a total ordering of the vertices but are restricted to the (random) edges of the graph. We extend this correspondence to infinite intervals, for which the situation is more intricate. We prove that almost surely every stable matching of the random bipartite graph obtained by performing Bernoulli percolation on the complete bipartite graph K_{Z,Z} falls into one of two classes: a countable family (σ_n)_{n∈Z} of tame stable matchings, in which the length of the longest edge crossing k is O(log|k|) as k→±∞, and an uncountable family of wild stable matchings, in which this length is expΩ(k) as k→+∞. The tame stable matching σn has the law of the Mallows permutation of Z (as constructed by Gnedin and Olshanski) composed with the shift k↦k+n. The permutation σn+1 dominates σn pointwise, and the two permutations are related by a shift along a random strictly increasing sequence.

  @article{1802.07142,
  Author = {Omer Angel and Alexander E. Holroyd and Tom Hutchcroft and Avi Levy},
  Title = {Mallows permutations as stable matchings},
  Year = {2018},
  Note = {arXiv:1802.07142},
  }
  

• 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 ]

  Let G be the product of finitely many trees T_1×T_2×⋯×T_N, each of which is regular with degree at least three. We consider Bernoulli bond percolation and the Ising model on this graph, giving a short proof that the model undergoes a second order phase transition with mean-field critical exponents in each case. The result concerning percolation recovers a result of Kozma (2013), while the result concerning the Ising model is new.

  We also present a new proof, using similar techniques, of a lemma of Schramm concerning the decay of the critical two-point function along a random walk, as well as some generalizations of this lemma.

  @article{1712.04911,
      AUTHOR = {Hutchcroft, Tom},
       TITLE = {Statistical physics on a product of trees},
     JOURNAL = {Ann. Inst. Henri Poincar\'{e} Probab. Stat.},
    FJOURNAL = {Annales de l'Institut Henri Poincar\'{e} Probabilit\'{e}s et
                Statistiques},
      VOLUME = {55},
        YEAR = {2019},
      NUMBER = {2},
       PAGES = {1001--1010},
        ISSN = {0246-0203},
         DOI = {10.1214/18-aihp906},
         URL = {https://doi.org/10.1214/18-aihp906},
  }
  

• Non-uniqueness and mean-field criticality for percolation on nonunimodular transitive graphs. Journal of the American Mathematical Society (JAMS), to appear.
  [ arXiv | pdf | abstract | BibTeX ]

  We study Bernoulli bond percolation on nonunimodular quasi-transitive graphs, and more generally graphs whose automorphism group has a nonunimodular quasi-transitive subgroup. We prove that percolation on any such graph has a non-empty phase in which there are infinite light clusters, which implies the existence of a non-empty phase in which there are infinitely many infinite clusters. That is, we show that p_c < p_h ≤ p_u for any such graph. This answers a question of Haggstrom, Peres, and Schonmann (1999), and verifies the nonunimodular case of a well-known conjecture of Benjamini and Schramm (1996).

  We also prove that the triangle condition holds at criticality on any such graph, which is known to imply that the critical exponents governing the percolation probability and the cluster volume take their mean-field values. Finally, we also prove that the susceptibility exponent, the gap exponent, and the cluster radius exponent each take their mean-field values on any such graph; these exponents were previously known to follow from the triangle condition only in the unimodular case.

  All our results apply, for example, to the product T_k×Z^d of a k-regular tree with Z^d for k≥3 and d≥1, for which these results were previously known only for large k. Furthermore, our methods also enable us to establish the basic topological features of the phase diagram for anisotropic percolation on such products, in which tree edges and Zd edges are given different retention probabilities. These features had only previously been established for d=1, k large

  @article{1711.02590,
  Author = {Tom Hutchcroft},
  Title = {Non-uniqueness and mean-field criticality for percolation on nonunimodular transitive graphs},
  Year = {2017},
  Note = {arXiv:1711.02590},
  }
  

• 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 ]

  We study the random planar map obtained from a critical, finite variance, Galton-Watson plane tree by adding the horizontal connections between successive vertices at each level. This random graph is closely related to the well-known causal dynamical triangulation that was introduced by Ambj{\o}rn and Loll and has been studied extensively by physicists. We prove that the horizontal distances in the graph are smaller than the vertical distances, but only by a subpolynomial factor: The diameter of the set of vertices at level n is both o(n) and n^(1−o(1)). This enables us to prove that the spectral dimension of the infinite version of the graph is almost surely equal to 2, and consequently that the random walk is diffusive almost surely. We also initiate an investigation of the case in which the offspring distribution is critical and belongs to the domain of attraction of an α-stable law for α∈(1,2), for which our understanding is much less complete.

  @article{1710.03137,
  Author = {Nicolas Curien and Tom Hutchcroft and Asaf Nachmias},
  Title = {Geometric and spectral properties of causal maps},
  Year = {2017},
  Note = {arXiv:1710.03137},
  }
  

• 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 ]

  We construct an example of a bounded degree, nonamenable, unimodular random rooted graph with pc=pu for Bernoulli bond percolation, as well as an example of a bounded degree, unimodular random rooted graph with pc<1 but with an infinite cluster at criticality. These examples show that two well-known conjectures of Benjamini and Schramm are false when generalised from transitive graphs to unimodular random rooted graphs.

  @article{1710.03003,
  Author = {Omer Angel and Tom Hutchcroft},
  Title = {Counterexamples for percolation on unimodular random graphs},
  Year = {2017},
  Note = {arXiv:1710.03003},
  }
  

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

  We study self-avoiding walk on graphs whose automorphism group has a transitive nonunimodular subgroup. We prove that self-avoiding walk is ballistic, that the bubble diagram converges at criticality, and that the critical two-point function decays exponentially in the distance from the origin. This implies that the critical exponent governing the susceptibility takes its mean-field value, and hence that the number of self-avoiding walks of length n is comparable to the nth power of the connective constant. We also prove that the same results hold for a large class of repulsive walk models with a self-intersection based interaction, including the weakly self-avoiding walk. All these results apply in particular to the product T_k×Z^d of a k-regular tree (k≥3) with Z^d, for which these results were previously only known for large k.

  @article{Hutch2017c,
      AUTHOR = {Hutchcroft, Tom},
       TITLE = {Self-avoiding walk on nonunimodular transitive graphs},
     JOURNAL = {Ann. Probab.},
    FJOURNAL = {The Annals of Probability},
      VOLUME = {47},
        YEAR = {2019},
      NUMBER = {5},
       PAGES = {2801--2829},
        ISSN = {0091-1798},
         DOI = {10.1214/18-AOP1322},
         URL = {https://doi.org/10.1214/18-AOP1322},
  }
  

• 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 ]

  The Hammersley-Welsh bound (1962) states that the number cn of length n self-avoiding walks on Z^d satisfies c_n≤exp[O(n^1/2)]μ_c^n, where μ_c=μ_c(d) is the connective constant of Z^d. While stronger estimates have subsequently been proven for d≥3, for d=2 this has remained the best rigorous, unconditional bound available. In this note, we give a new, simplified proof of this bound, which does not rely on the combinatorial analysis of unfolding. We also prove a small, non-quantitative improvement to the bound, namely c_n≤exp[o(n^1/2)]μ_c^n. The improved bound is obtained as a corollary to the sub-ballisticity theorem of Duminil-Copin and Hammond (2013). We also show that any quantitative form of that theorem would yield a corresponding quantitative improvement to the Hammersley-Welsh bound.

  @article{Hutch2017b,
  author = "Hutchcroft, Tom",
  doi = "10.1214/17-ECP94",
  fjournal = "Electronic Communications in Probability",
  journal = "Electron. Commun. Probab.",
  pages = "8 pp.",
  pno = "5",
  publisher = "The Institute of Mathematical Statistics and the Bernoulli Society",
  title = "The Hammersley-Welsh bound for self-avoiding walk revisited",
  url = "https://doi.org/10.1214/17-ECP94",
  volume = "23",
  year = "2018"
  }
  
  

• 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 ]

  We construct stationary finitely dependent colorings of the cycle which are analogous to the colorings of the integers recently constructed by Holroyd and Liggett. These colorings can be described by a simple necklace insertion procedure, and also in terms of an Eden growth model on a tree. Using these descriptions we obtain simpler and more direct proofs of the characterizations of the 1- and 2-color marginals.

  @article{HoHuLe2,
  author = "Holroyd, Alexander E. and Hutchcroft, Tom and Levy, Avi",
  doi = "10.1214/18-ECP118",
  fjournal = "Electronic Communications in Probability",
  journal = "Electron. Commun. Probab.",
  pages = "12 pp.",
  pno = "64",
  publisher = "The Institute of Mathematical Statistics and the Bernoulli Society",
  title = "Finitely dependent cycle coloring",
  url = "https://doi.org/10.1214/18-ECP118",
  volume = "23",
  year = "2018"
  }
  
  

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

  Benjamini and Schramm (1996) used circle packing to prove that every transient, bounded degree planar graph admits non-constant harmonic functions of finite Dirichlet energy. We refine their result, showing in particular that for every transient, bounded degree, simple planar triangulation T and every circle packing of T in a domain D, there is a canonical, explicit bounded linear isomorphism between the space of harmonic Dirichlet functions on T and the space of harmonic Dirichlet functions on D.

  @article{Hutch2017a,
      AUTHOR = {Hutchcroft, Tom},
       TITLE = {Harmonic {D}irichlet functions on planar graphs},
     JOURNAL = {Discrete Comput. Geom.},
    FJOURNAL = {Discrete \& Computational Geometry. An International Journal
                of Mathematics and Computer Science},
      VOLUME = {61},
        YEAR = {2019},
      NUMBER = {3},
       PAGES = {479--506},
        ISSN = {0179-5376},
         DOI = {10.1007/s00454-019-00057-2},
         URL = {https://doi.org/10.1007/s00454-019-00057-2},
  }
  

• Mallows Permutations and Finite Dependence, with Alexander E. Holroyd and Avi Levy. Annals of Probability, to appear.
  [ arXiv | pdf | abstract | BibTeX ]

  We use the Mallows permutation model to construct a new family of stationary finitely dependent proper colorings of the integers. We prove that these colorings can be expressed as finitary factors of i.i.d. processes with finite mean coding radii. They are the first colorings known to have these properties. Moreover, we prove that the coding radii have exponential tails, and that the colorings can also be expressed as functions of countable-state Markov chains. We deduce analogous existence statements concerning shifts of finite type and higher-dimensional colorings.

  @article{HoHuLe1,
    title={Mallows Permutations and Finite Dependence},
    author={Holroyd, Alexander E. and Hutchcroft, Tom
    and Levy, Avi},
    journal={arXiv preprint},
    note={arXiv:1706.09526}
    year={2017}
  }
  

• Indistinguishability of collections of trees in the uniform spanning forest. Annales de l’Institut Henri Poincaré (AIHP), to appear.
  [ arXiv | pdf | abstract | BibTeX ]

  We prove the following indistinguishability theorem for k-tuples of trees in the uniform spanning forest of ℤ^d: Suppose that 𝒜 is a property of a k-tuple of components that is stable under finite modifications of the forest. Then either every k-tuple of distinct trees has property 𝒜 almost surely, or no k-tuple of distinct trees has property 𝒜 almost surely. This generalizes the indistinguishability theorem of the author and Nachmias (2016), which applied to individual trees. Our results apply more generally to any graph that has the Liouville property and for which every component of the USF is one-ended.

  @article{1810.06382,
    title={Indistinguishability of collections of trees in the uniform spanning forest},
    author={Hutchcroft, Tom},
    journal={arXiv preprint arXiv:1810.06382},
    year={2018}
  }
  

• 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 ]

  We prove that the uniform spanning forests of Z^d and Z^l have qualitatively different connectivity properties whenever l > d ≥ 4. In particular, we consider the graph formed by contracting each tree of the uniform spanning forest down to a single vertex, which we call the component graph. We show that the set of ubiquitous subgraphs of the component graph changes whenever the dimension changes and is above 8. To separate dimensions 5, 6, 7, and 8, we prove a similar result concerning ubiquitous subhypergraphs in the component hypergraph. Our result sharpens a theorem of Benjamini, Kesten, Peres, and Schramm, who proved that the diameter of the component graph increases by one every time the dimension increases by four.

  @article{HutPe17,
      AUTHOR = {Hutchcroft, Tom and Peres, Yuval},
       TITLE = {The component graph of the uniform spanning forest:
                transitions in dimensions {$9,10,11,\ldots $}},
     JOURNAL = {Probab. Theory Related Fields},
    FJOURNAL = {Probability Theory and Related Fields},
      VOLUME = {175},
        YEAR = {2019},
      NUMBER = {1-2},
       PAGES = {141--208},
        ISSN = {0178-8051},
         DOI = {10.1007/s00440-018-0884-3},
         URL = {https://doi.org/10.1007/s00440-018-0884-3},
  }
  

• 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 ]

  We show that for infinite planar unimodular random rooted maps, many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply connected unimodular random rooted map is sofic, that is, a Benjamini-Schramm limit of finite maps.

  @article {MR3820434,
      AUTHOR = {Angel, Omer and Hutchcroft, Tom and Nachmias, Asaf and Ray,
                Gourab},
       TITLE = {Hyperbolic and {P}arabolic {U}nimodular {R}andom {M}aps},
     JOURNAL = {Geom. Funct. Anal.},
    FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {28},
        YEAR = {2018},
      NUMBER = {4},
       PAGES = {879--942},
        ISSN = {1016-443X},
         DOI = {10.1007/s00039-018-0446-y},
         URL = {https://doi.org/10.1007/s00039-018-0446-y},
         EPRINT={arXiv:1612.08693}
  }
  

• 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 ]

  We prove that critical percolation on any quasi-transitive graph of exponential volume growth does not have a unique infinite cluster. This allows us to deduce from earlier results that critical percolation on any graph in this class does not have any infinite clusters. The result is new when the graph in question is either amenable or nonunimodular.

  @article{Hutchcroft2016944,
  title = "Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters ",
  journal = "Comptes Rendus Mathematique ",
  volume = "354",
  number = "9",
  pages = "944 - 947",
  year = "2016",
  note = "",
  issn = "1631-073X",
  doi = "http://dx.doi.org/10.1016/j.crma.2016.07.013",
  url = "http://www.sciencedirect.com/science/article/pii/S1631073X16301352",
  author = "Tom Hutchcroft"
  }
  

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

  We prove that the free uniform spanning forest of any bounded degree, proper plane graph is connected almost surely, answering a question of Benjamini, Lyons, Peres and Schramm. We provide a quantitative form of this result, calculating the critical exponents governing the geometry of the uniform spanning forests of transient proper plane graphs with bounded degrees and codegrees. We find that these exponents are universal in this class of graphs, provided that measurements are made using the hyperbolic geometry of their circle packings rather than their usual combinatorial geometry.

  @article{HutNach2016b,
      AUTHOR = {Hutchcroft, Tom and Nachmias, Asaf},
       TITLE = {Uniform spanning forests of planar graphs},
     JOURNAL = {Forum Math. Sigma},
    FJOURNAL = {Forum of Mathematics. Sigma},
      VOLUME = {7},
        YEAR = {2019},
       PAGES = {e29, 55},
         DOI = {10.1017/fms.2019.14},
         URL = {https://doi.org/10.1017/fms.2019.14},
  }
  

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

  We extend the Aldous-Broder algorithm to generate the wired uniform spanning forests (WUSFs) of infinite, transient graphs. We do this by replacing the simple random walk in the classical algorithm with Sznitman's random interlacement process. We then apply this algorithm to study the WUSF, showing that every component of the WUSF is one-ended almost surely in any graph satisfying a certain weak anchored isoperimetric condition, that the number of `excessive ends' in the WUSF is non-random in any graph, and also that every component of the WUSF is one-ended almost surely in any transient unimodular random rooted graph. The first two of these results answer positively two questions of Lyons, Morris and Schramm, while the third extends a recent result of the author.

  Finally, we construct a counterexample showing that almost sure one-endedness of WUSF components is not preserved by rough isometries of the underlying graphs, answering negatively a further question of Lyons, Morris and Schramm.

  @article{hutchcroft2018,
  author = "Hutchcroft, Tom",
  doi = "10.1214/17-AOP1203",
  fjournal = "The Annals of Probability",
  journal = "Ann. Probab.",
  month = "03",
  number = "2",
  pages = "1170--1200",
  publisher = "The Institute of Mathematical Statistics",
  title = "Interlacements and the wired uniform spanning forest",
  url = "https://doi.org/10.1214/17-AOP1203",
  volume = "46",
  year = "2018"
  }
    

• 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 ]

  We give a new proof that the Poisson boundary of a planar graph coincides with the boundary of its square tiling and with the boundary of its circle packing, originally proven by Georgakopoulos and Angel, Barlow, Gurel-Gurevich and Nachmias respectively. Our proof is robust, and also allows us to identify the Poisson boundaries of graphs that are rough-isometric to planar graphs.

  We also prove that the boundary of the square tiling of a bounded degree plane triangulation coincides with its Martin boundary. This is done by comparing the square tiling of the triangulation with its circle packing.

  @article{hutchcroft2015boundaries,
    title={Boundaries of Planar Graphs: A Unified Approach},
    author={Hutchcroft, Tom and Peres, Yuval},
    journal={arXiv preprint arXiv:1508.03923},
    year={2015}
  }
  

• 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 ]

  We prove that in both the free and the wired uniform spanning forest (FUSF and WUSF) of any unimodular random rooted network (in particular, of any Cayley graph), it is impossible to distinguish the connected components of the forest from each other by invariantly defined graph properties almost surely. This confirms a conjecture of Benjamini, Lyons, Peres and Schramm.

  We use this to answer positively two additional questions of Benjamini, Lyons, Peres and Schramm under the assumption of unimodularity. We prove that on any unimodular random rooted network, the FUSF is either connected or has infinitely many connected components almost surely, and, if the FUSF and WUSF are distinct, then every component of the FUSF is transient and infinitely-ended almost surely. All of these results are new even for Cayley graphs.

  @Article{HutNach2016a,
  author="Hutchcroft, Tom
  and Nachmias, Asaf",
  title="Indistinguishability of trees in uniform spanning forests",
  journal="Probability Theory and Related Fields",
  year="2016",
  pages="1--40",
  issn="1432-2064",
  doi="10.1007/s00440-016-0707-3",
  url="http://dx.doi.org/10.1007/s00440-016-0707-3"
  }
  

• 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 ]

  We prove that in any recurrent reversible random rooted graph, two independent simple random walks started at the same vertex collide infinitely often almost surely. This applies to the Uniform Infinite Planar Triangulation and Quadrangulation and to the Incipient Infinite Cluster in Z^2.

  @article{HutPe2015a,
  author = "Hutchcroft, Tom and Peres, Yuval",
  doi = "10.1214/ECP.v20-4330",
  fjournal = "Electronic Communications in Probability",
  journal = "Electron. Commun. Probab.",
  pages = "6 pp.",
  pno = "63",
  publisher = "The Institute of Mathematical Statistics and the Bernoulli Society",
  title = "Collisions of random walks in reversible random graphs",
  url = "http://dx.doi.org/10.1214/ECP.v20-4330",
  volume = "20",
  year = "2015"
  }
  

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

  We prove that every component of the wired uniform spanning forest (WUSF) is one-ended almost surely in every transient reversible random graph, removing the bounded degree hypothesis required by earlier results. We deduce that every component of the WUSF is one-ended almost surely in every supercritical Galton-Watson tree, answering a question of Benjamini, Lyons, Peres and Schramm.

  Our proof introduces and exploits a family of Markov chains under which the oriented WUSF is stationary, which we call the wired cycle-breaking dynamics.

  @article {H15a,
      AUTHOR = {Hutchcroft, Tom},
       TITLE = {Wired cycle-breaking dynamics for uniform spanning forests},
     JOURNAL = {Ann. Probab.},
    FJOURNAL = {The Annals of Probability},
      VOLUME = {44},
        YEAR = {2016},
      NUMBER = {6},
       PAGES = {3879--3892},
        ISSN = {0091-1798},
         DOI = {10.1214/15-AOP1063},
         URL = {http://dx.doi.org/10.1214/15-AOP1063},
  }
    
  

• 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 ]

  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 amenable in the sense of Aldous and Lyons. As a part of this, we obtain an alternative proof of the Benjamini-Schramm Recurrence Theorem.

  Secondly, in the hyperbolic case, we prove that the random walk almost surely converges to a point in the unit circle, that the law of this limiting point has full support and no atoms, and that the unit circle is a realisation of the Poisson boundary. Finally, we show that the simple random walk has positive speed in the hyperbolic metric.

  @Article{AHNR2016,
  author="Angel, Omer
  and Hutchcroft, Tom
  and Nachmias, Asaf
  and Ray, Gourab",
  title="Unimodular hyperbolic triangulations: circle packing and random walk",
  journal="Inventiones mathematicae",
  year="2016",
  volume="206",
  number="1",
  pages="229--268",
  issn="1432-1297",
  doi="10.1007/s00222-016-0653-9",
  url="http://dx.doi.org/10.1007/s00222-016-0653-9"
  }
  

Other Writing

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

Awards and Honours

• Rollo Davidson Prize, 2019.
• Canadian Mathematical Society Doctoral Prize, 2018.
• Governor General's Gold Medal, UBC, 2018.
• Microsoft Research PhD Fellowship, 2016.