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Random Trees in Hyperbolic FPP

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Pith's one-line read First-passage percolation geodesics on a hyperbolic group coalesce so quickly that the random merge point stays within N of the ideal triangle's centroid with probability at least 1 - e^{-cN}.

desk verdict A useful continuation of the authors' program with real new outputs, but the exponential tail rests on imported 1-dependence claims and the continuity theorem has a visible gap; both are fixable but not yet closed. read the letter →

arxiv 2607.25567 v1 pith:SPJ4EWKQ submitted 2026-07-28 math.PR math.GRmath.GT

Riddhipratim Basu , Mahan Mj This is my paper · ORCID
classification math.PRmath.GRmath.GT MSC 60K3582B4320F6720F6551F9960J50
keywords firstpassagepercolationhyperbolicgroupcoalescencegeodesictreeGromovboundarybackwardcentroidexponentialtail
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies first passage percolation on Cayley graphs of Gromov-hyperbolic groups, where each edge gets an independent random length and geodesics are shortest paths in the resulting random metric. Earlier work of the same authors showed that all geodesic rays pointing to a fixed boundary point almost surely merge, forming a random tree. The new results here make this quantitative: the point where two geodesics coalesce is exponentially unlikely to be far from the deterministic centroid of the ideal triangle formed by the two starting boundary points and the target. This exponential tail is then shown to control the geometry of the backward tree: it yields separation of subtrees rooted on a horosphere, identifies the average growth rate of the tree, proves continuity of the tree law as a function of the boundary triple, and produces finite dead-end subtrees under an additional hypothesis. A sympathetic reader should care because a qualitative coalescence phenomenon becomes a precise tool, opening the way to exact statistics of random geodesic trees in negative curvature.

What carries the argument

The load-bearing mechanism is the slab-coalescence estimate: for a fixed boundary direction ξ and a sufficiently large scale D, earlier work supplies events B_i of uniform positive probability, depending only on the environment in a slab between two hyperplanes, on which every pair of geodesic rays in direction ξ intersects inside the central geodesic segment of the slab. Because D can be chosen so these events are 1-dependent in i, the probability that a pair of rays fails to coalesce across n consecutive slabs decays exponentially in n (Theorem 4.5). Theorem 4.2 combines this with hyperplanes transverse to the three sides of the ideal triangle, bounding the probability that the coalescence

What would settle it

Simulate FPP with exponential edge weights on a large ball of a surface-group Cayley graph, fix boundary points a, b, ξ, and compute the empirical distribution of the distance X_{abξ} between the deterministic and random centroids. If this tail decays more slowly than exponentially (e.g., polynomially) in N for any fixed triple, Theorem 4.2 is false; a deviation of the measured average growth rate of the backward tree from log λ would also indicate the concentration argument overshoots.

Watch

Extended reading notes

Core claim

The central discovery is the exponential tail estimate (Theorem 4.2): for any three distinct boundary points a, b, ξ, the distance X_{abξ} between the deterministic centroid of the ideal triangle (a,b,ξ) and the random coalescence point of the two random geodesics directed at ξ satisfies P[X_{abξ} ≥ N] ≤ e^{-cN} for all large N, with c>0 uniform over all boundary triples. The proof decomposes the triangle into hyperplane-bounded slabs and uses slab events of uniformly positive probability to show that coalescence is forced to happen quickly in each slab, so that backtracking or running far from the centroid is exponentially unlikely. This estimate is treated as the load-bearing input: it imp

Load-bearing premise

The chain rests on an imported quantitative coalescence estimate — that for some fixed scale, slab events forcing all geodesics in a direction to intersect have uniformly positive probability and can be made 1-dependent — and the proof of the derived exponential decay is only sketched; if that estimate fails or its constants degrade, the exponential tail and everything built on it does not follow.

Editorial extensions

If this is right

  • The coalescence-centroid distance has an exponential tail uniformly over all boundary triples, so the random tree's branch points are exponentially concentrated around the deterministic centroids.
  • Backward subtrees rooted at points on a horosphere that are far apart in the horosphere metric are disjoint with overwhelming probability, and in the group metric up to a superpolynomial error.
  • The average growth rate of the backward tree exists and equals log λ, where λ is the Perron-Frobenius eigenvalue of the finite-state automaton generating the geodesic language of the group.
  • The law of the backward tree built from the random centroid is continuous in the boundary triple modulo the group action; if an asymptotic law exists for almost every boundary pair, ergodicity forces it to be constant.
  • Under an additional positive-density hypothesis on the edge-weight law, with positive probability there are finite backward subtrees — 'bubbles of positive curvature' — so the FPP metric is almost surely not Gromov-hyperbolic, reproducing a known non-hyperbolicity result under more restrictive assumptions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The exponential tail likely implies that Busemann functions for the FPP metric have exponential concentration along geodesic rays, a quantitative strengthening of the linear variance growth known from earlier work.
  • The continuity of the tree law on boundary triples suggests the existence of a canonical invariant random tree law for the FPP; a natural test is whether the limiting law is the same for any two triples related by the group action.
  • The bubble construction requires a positive density of edge weights near zero; it is plausible that dead-ends occur under the mere super-exponential-tail assumption, which could be tested by simulation on a surface group with exponential weights.
  • The growth-rate identity log λ may be interpreted as a 'dimension' of the random tree; comparing it with the Hausdorff dimension of the boundary could yield further exact identities for hyperbolic FPP trees.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This paper studies first passage percolation (FPP) on Cayley graphs of Gromov-hyperbolic groups with i.i.d. continuous edge weights having super-exponential tails. It first surveys earlier results, chiefly from the authors' papers [BM22, BM26], on coalescence of FPP geodesics and on the associated backward random tree T(ξ,ω). The new results are: an exponential tail for the distance between the random coalescence point of two geodesics directed at a boundary point and the deterministic centroid of the ideal triangle (Theorem 4.2); consequences for disjoint cones and for the average growth rate of backward trees, which is identified as log λ (Theorem 5.11); under an added support condition on the edge-length distribution, existence of finite backward subtrees and of 'bubbles of positive curvature', yielding non-hyperbolicity of the random metric (Section 5.3, Corollaries 5.15, 5.16); and continuity of the law of backward subtrees rooted at the random centroid as a function of the boundary triple (Theorem 5.21).

Significance. The exponential coalescence estimate in Theorem 4.2 is a genuine quantitative step beyond the earlier almost-sure coalescence results and, if correct, is likely to be useful for further study of hyperbolic FPP. The identification of the average growth rate with log λ and the construction of finite backward subtrees are natural and interesting. The paper also serves a useful expository role. However, the central estimate depends on a theorem whose proof is only sketched and on an imported proposition; one of the main later theorems contains a visible gap in the logic of its proof. These issues need to be addressed before the results can be considered established.

major comments (4)
  1. [§4.1, Theorem 4.5] Theorem 4.5 is a new effective coalescence/backtracking estimate and is the main input to Theorem 4.2, yet its proof is dismissed with 'we omit the details'. In particular, the assertion that the slab events B_i from Proposition 4.3 can be made 1-dependent is not demonstrated. The supports N_{D/10}(S(i,D)) are disjoint for indices differing by at least 2 only if the hyperplanes are exactly placed; the coarse nature of hyperplanes is exactly what the sentence 'these events are not disjoint since the hyperplanes are only coarsely well defined' refers to. The 1-dependence is essential for the product estimate (1−β)^n that gives the exponential rate. Since Theorem 4.2 and everything downstream (Corollary 5.3, Theorem 5.11, Theorem 5.21) rely on this, the paper must either provide a complete proof of Theorem 4.5, including the 1-dependence verification, or cite a precise theorem in [BM22] tha
  2. [§5.4, Theorem 5.21] The proof of continuity of L(x,y,z) has a visible gap. It is asserted that if [x_n,z]_ω and [x,z]_ω coalesce within distance N of C_n with high probability, then the backward trees rooted at c(x_n,y,z,ω) and c(x,y,z,ω) 'coincide on the nose' on a set of probability at least 1−e^{−cN}. Pairwise coalescence of two z-directed geodesics does not by itself imply that the random centroids c(x_n,y,z,ω) and c(x,y,z,ω) are equal, nor that the entire rooted backward trees coincide: the coalescence of [x_n,z]_ω and [x,z]_ω says nothing about where either merges with [y,z]_ω. Moreover, the displayed definition 'X_n equal to the distance of the random centroid of 1, x_n, x from C_n' is not the quantity that would control coalescence of [x_n,z] and [x,z]; the role of '1' is unclear. A correct proof would need a stronger coupling of the finite subtrees of T(ξ,ω), not merely pairwise coalescence of two
  3. [§4.2, proof of Theorem 4.2] Theorem 4.5(1) is stated for starting points o1, o2 ∈ G, but the proof of Theorem 4.2 applies it to boundary points a, b ∈ ∂G. The opening sentence says one may assume sequences a_n → a, b_n → b without loss of generality, but no limiting argument is supplied. Since the exact rate and uniformity in (a,b,ξ) are the point of the theorem, this passage from finite vertices to boundary points needs a careful justification. This may be routine, but as written it is another omitted technical step in the central estimate.
  4. [§5.3, Proposition 5.14] The additional hypothesis that the edge-length distribution ρ has support containing an interval [0,h] is introduced in this section and is stronger than Assumption 2.3. It should be flagged as a new standing assumption wherever Theorem 5.11's successors or Corollaries 5.15 and 5.16 are stated. More substantively, the proof that a is a dead-end is incomplete: it rules out [a,ξ)_ω ⊂ [b,ξ)_ω only for b outside B_R(a), but a could still be a proper sub-ray of [b,ξ)_ω for some b inside B_R(a). The bubble contradiction can be made to cover this case as well, but the argument as written does not.
minor comments (5)
  1. [§4.3, Proposition 4.8] In the first case, the displayed total variation distance between the joint distribution of (c_{a1ξ}(ω), c_{a1ξ}(ω)) should presumably be between c_{a1ξ}(ω) and c_{b1ξ}(ω).
  2. [§5.4, Theorem 5.21] The phrase 'random centroid of 1, x_n, x' is confusing because 1 is not a point of ∂G. Please clarify the intended triple.
  3. [§5.2, Definition 5.10] The averaging over the Følner sequence L_k is written informally with an integral and limsup. Since this definition is used in Theorem 5.11, it would help to make the limiting procedure fully precise.
  4. [§6.1] The toy example for G=Z uses a Dirac mass at 1, which violates the standing assumption that ρ has no atoms. The example is explicitly heuristic, but it should be labelled as outside the main hypotheses.
  5. [References] [CG25] is cited as 'personal communication'; if it is now a preprint, an arXiv identifier or journal reference should be given.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found: central results are derived from (not equivalent to) the authors' prior [BM22]/[BM26] estimates; the main issue is an omitted proof of Theorem 4.5, a verification gap rather than circularity.

full rationale

The paper is explicitly partly expository and builds on [BM22, BM26]. The load-bearing Theorem 4.2 (exponential tail of X_{abξ}) is proved from Theorem 4.5 and Proposition 3.7. Theorem 4.5 is a quantitative version of [BM22, Prop 7.10] with proof omitted: 'Since the arguments are quite similar to the ones there, we omit the details.' The essential 1-dependence of the slab events is asserted rather than proved. This is a reliance on same-author prior work and an unverified premise, but not a circular reduction: Proposition 4.3 is a theorem with stated assumptions (continuous ρ, superexponential tails) that do not include X_{abξ}; no parameter is fitted to X_{abξ} and renamed as a prediction. Theorem 5.11 derives the average growth rate log λ from the deterministic growth of C_D(R) (Prop 5.8) and the probabilistic mass estimate (Prop 5.9); the definition of average growth rate does not contain λ, so the result is not self-definitional. Corollary 5.16 explicitly credits [BJQ25, Theorem A] and gives an independent bubble construction. No equation in the derivation chain is equivalent to its input by construction. Score 2 reflects the heavy load-bearing use of the authors' own prior estimates and the omitted proof of Theorem 4.5, not a detected circular step.

Assumptions & free parameters 0 free parameters · 9 assumptions · 1 invented entities

No empirical data or fitted constants appear; the model is a family (G,ρ) of i.i.d. edge weights. Auxiliary proof constants such as the scale D, ϵ, and k=D^α are chosen to make arguments work, but they do not enter the theorem statements as free parameters. The axioms above are the mathematical assumptions on which the new results rest; the most load-bearing is the imported slab-coalescence estimate [BM22, Prop 7.10].

assumptions (9)
  • domain assumption Assumption 2.3: edge weights are i.i.d., non-negative, continuous (no atoms), with super-exponential tail: ∫_0^∞ e^{a x} dρ(x) < ∞ for all a>0.
    Defines the FPP model; used throughout for uniqueness of geodesics and for exponential estimates in Sections 3-5.
  • standard math G is a Gromov-hyperbolic group with a fixed finite symmetric generating set; the Cayley graph Γ is a proper Gromov-hyperbolic graph; the Patterson-Sullivan measure ν on ∂G is used for boundary averaging.
    The whole paper works in this setting; existence and basic properties of ν are cited to [Coo93] and [Cal13].
  • standard math Cannon's automatic structure: the geodesic language of G is regular, accepted by a finite automaton with Perron-Frobenius eigenvalue λ and maximal components; all but exponentially few long geodesics lie in a maximal component.
    Used in Corollary 2.6, Proposition 2.8, and Corollary 5.7 to control the volume of geodesic shadows Θ(g,R,n,m).
  • standard math The G-action on (∂G,ν) is ergodic, and the G-action on (∂²G,ν_BMS) is ergodic (Bader-Furman).
    Used to conclude v(ξ) is a.e. constant (Theorem 2.5) and to conclude that an existing asymptotic law L(z',z) is a.e. constant (Proposition 5.19).
  • domain assumption Proposition 4.3 from [BM22, Prop 7.10]: for sufficiently large D, slab events B_i have uniform positive probability β and ensure that all ω-geodesics from arbitrary starting points to ξ intersect on the slab segment; these events can be chosen 1-dependent.
    This imported quantitative coalescence estimate is the foundation of Theorem 4.5 and Theorem 4.2. It is the most fragile premise; the proof is not reproduced in this paper.
  • standard math Uniformly quasiconvex hyperplanes perpendicular to geodesics exist, and nearest-point projection estimates (Propositions 3.6 and 3.7 from [BM26]) hold.
    Used in the proof of Theorem 4.2 and in Proposition 2.10; these are cited as established results from the authors' prior work.
  • domain assumption One-endedness of G, giving a uniform k such that any two points in the sphere S_C(x) are joined by a path in the k-shell (Bestvina-Mess).
    Assumed explicitly in Section 5.3 for the bubble-of-positive-curvature construction and dead-end results.
  • ad hoc to paper Additional hypothesis in Section 5.3: the edge-length distribution ρ has support containing an interval of the form [0,h].
    Stronger than Assumption 2.3 and used specifically to force the bubble event with positive probability in Proposition 5.14.
  • standard math Horospheres in a one-ended hyperbolic group are exponentially distorted and (L,d_L) has polynomial growth (Lemma 5.4, cited from [LMM24]).
    Used to convert exponential graph-distance decay into superpolynomial decay in the intrinsic horosphere metric in Theorem 5.5.
invented entities (1)
  • Bubble of positive curvature (Definition 5.12)
    purpose: A ball in which random geodesics between boundary points must stay in a thin shell; used to prove existence of dead-ends and non-hyperbolicity of (Γ,d_ω).
    This is a new mathematical definition introduced in the paper; it has no empirical or falsifiable handle outside the model. Its role is internal to the proof, not an independently testable prediction.

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Pith. "Pith review of Random Trees in Hyperbolic FPP." pith.science (2026). https://pith.science/paper/SPJ4EWKQ

@misc{pith2026260725567,
  author       = {Pith},
  title        = {Pith review of: Random Trees in Hyperbolic FPP},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SPJ4EWKQ}},
  note         = {Machine review of arXiv:2607.25567}
}
abstract

We start by surveying known properties of first passage percolation (FPP) geodesics on a Gromov-hyperbolic group $G$. Due to a coalescence phenomenon established in earlier work of the authors, a random tree $T(\xi,\omega)$ consisting of infinite random geodesics to a point $\xi$ on the Gromov boundary $\partial G$ emerges naturally. These trees have a rich random geometry that we explore further in this paper.

Figures

Figures reproduced from arXiv: 2607.25567 by the authors.

Figure 1
Figure 1. Proof of Theorem 4.2: We want to show that the coa￾lescence point z of the geodesics (a, ξ)ω and (b, ξ)ω cannot be too far from the deterministic centroid y. By considering hyperplanes normal to [y, a), [y, b) and [y, ξ) and using Theorem 4.5 we know that z is unlikely to lie beyond too many of these hyperplanes from y in any of the three directions. That z is typically close to y is then deduced using Proposition 3… view at source ↗
Figure 2
Figure 2. Proof of the first case in Proposition 4.8: the points b1, b2, b3 are at distance N/4 from πξ(b) on [1, πξ(b)], [b, πξ(b)] and [πξ(a), πξ(b)] respectively. Points b ′ i (not marked in the figure) are midpoints of [bi , πξ(b)] and Hb ′ i are the hyperplanes passing through b ′ i perpendicular to these line segments. We show that with high probability the coalescence points cb1ξ(ω) equals a ran￾dom point cb1b2b3 (ω) w… view at source ↗

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Works this paper leans on

38 extracted references · 4 linked inside Pith

  1. [1]

    50 years of first-passage percolation , volume 68 of University Lecture Series

    Antonio Auffinger, Michael Damron, and Jack Hanson. 50 years of first-passage percolation , volume 68 of University Lecture Series . American Mathematical Society, Providence, RI, 2017

    2017

  2. [2]

    Ahlberg and C

    D. Ahlberg and C. Hoffman. Random coalescing geodesics in first-passage percolation. Preprint, arXiv:1609.02447

    arXiv

  3. [3]

    Geodesic trees in last passage percolation and some related problems

    M\'arton Bal\'azs, Riddhipratim Basu, and Sudeshna Bhattacharjee. Geodesic trees in last passage percolation and some related problems. arXiv preprint arXiv:2308.07312 , 2023

    arXiv 2023

  4. [4]

    Bader and A

    U. Bader and A. Furman. Some ergodic properties of metrics on hyperbolic groups . preprint, arXiv:1707.02020 , page 23pp., 2017

    arXiv 2017

  5. [5]

    Greenberg- S halom's commensurator arithmeticity hypothesis and applications

    Nic Brody, David Fisher, Mahan Mj, and Wouter van Limbeek. Greenberg- S halom's commensurator arithmeticity hypothesis and applications. preprint, ar X iv:2308.07785 , 2023

    arXiv 2023

  6. [6]

    First-passage percolation, non-positive curvature, and radial maps, 2025

    Dominic Bair, Sagnik Jana, and Yulan Qing. First-passage percolation, non-positive curvature, and radial maps, 2025

    2025

  7. [7]

    The boundary of negatively curved groups

    Mladen Bestvina and Geoffrey Mess. The boundary of negatively curved groups. J. Amer. Math. Soc. , 4(3):469--481, 1991

    1991

  8. [8]

    First passage percolation on hyperbolic groups

    Riddhipratim Basu and Mahan Mj. First passage percolation on hyperbolic groups. Adv. Math. , 408(part A):Paper No. 108599, 63, 2022

    2022

Show all 38 references
  1. [9]

    Geodesic trees and exceptional directions in fpp on hyperbolic groups

    Riddhipratim Basu and Mahan Mj. Geodesic trees and exceptional directions in fpp on hyperbolic groups. Advances in Mathematics , 498:111030, 2026

    2026

  2. [10]

    B. H. Bowditch. A topological characterization of hyperbolic groups. J. Amer. Math. Soc. 11 , pages 643--667, 1998

    1998

  3. [11]

    First passage percolation on a hyperbolic graph admits bi-infinite geodesics

    Itai Benjamini and Romain Tessera. First passage percolation on a hyperbolic graph admits bi-infinite geodesics. Electron. Commun. Probab. , 22:Paper No. 14, 8, 2017

    2017

  4. [12]

    Tightness of fluctuations of first passage percolation on some large graphs

    Itai Benjamini and Ofer Zeitouni. Tightness of fluctuations of first passage percolation on some large graphs. In Geometric aspects of functional analysis , volume 2050 of Lecture Notes in Math. , pages 127--132. Springer, Heidelberg, 2012

  5. [13]

    The ergodic theory of hyperbolic groups

    Danny Calegari. The ergodic theory of hyperbolic groups. In Geometry and topology down under , volume 597 of Contemp. Math. , pages 15--52. Amer. Math. Soc., Providence, RI, 2013

    2013

  6. [14]

    James W. Cannon. The combinatorial structure of cocompact discrete hyperbolic groups. Geom. Dedicata , 16(2):123--148, 1984

    1984

  7. [15]

    James W. Cannon. The theory of negatively curved spaces and groups. In Ergodic theory, symbolic dynamics, and hyperbolic spaces ( T rieste, 1989) , Oxford Sci. Publ., pages 315--369. Oxford Univ. Press, New York, 1991

    1989

  8. [16]

    Amenable hyperbolic groups

    Pierre-Emmanuel Caprace, Yves Cornulier, Nicolas Monod, and Romain Tessera. Amenable hyperbolic groups. J. Eur. Math. Soc. (JEMS) , 17(11):2903--2947, 2015

    2015

  9. [17]

    Combable functions, quasimorphisms, and the central limit theorem

    Danny Calegari and Koji Fujiwara. Combable functions, quasimorphisms, and the central limit theorem. Ergodic Theory Dynam. Systems , 30(5):1343--1369, 2010

    2010

  10. [18]

    Thick trace at infinity for the hyperbolic radial spanning tree, 2023

    David Coupier, Lucas Flammant, and Viet Chi Tran. Thick trace at infinity for the hyperbolic radial spanning tree, 2023

    2023

  11. [19]

    personal communication

    Kunal Chawla and Christian Gorski, 2025. personal communication

    2025

  12. [20]

    Statistics and compression of scl

    Danny Calegari and Joseph Maher. Statistics and compression of scl. Ergodic Theory Dynam. Systems , 35(1):64--110, 2015

    2015

  13. [21]

    Coornaert

    M. Coornaert. Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov . Pacific J. Math. 159, no. 2 , pages 241--270, 1993

    1993

  14. [22]

    Multiple geodesics with the same direction

    David Coupier. Multiple geodesics with the same direction. Electron. Commun. Probab. , 16:517--527, 2011

    2011

  15. [23]

    Cannon and W

    J. Cannon and W. P. Thurston. Group I nvariant P eano C urves. preprint, Princeton , 1985

    1985

  16. [24]

    Cannon and W

    J. Cannon and W. P. Thurston. Group I nvariant P eano C urves. Geom. Topol. 11 , pages 1315--1355, 2007

    2007

  17. [25]

    Busemann functions and infinite geodesics in two-dimensional first-passage percolation

    Michael Damron and Jack Hanson. Busemann functions and infinite geodesics in two-dimensional first-passage percolation. Comm. Math. Phys. , 325(3):917--963, 2014

    2014

  18. [26]

    Bigeodesics in first-passage percolation

    Michael Damron and Jack Hanson. Bigeodesics in first-passage percolation. Communications in Mathematical Physics , 349(2):753--776, 2017

    2017

  19. [27]

    Ghys and P

    E. Ghys and P. de la Harpe(eds.). Sur les groupes hyperboliques d'apr\'es M ikhael G romov. Progress in Math. vol 83, Birkh \" a user, Boston, MA . , 1990

    1990

  20. [28]

    Coexistence for R ichardson type competing spatial growth models

    Christopher Hoffman. Coexistence for R ichardson type competing spatial growth models. Ann. Appl. Probab. , 15(1B):739--747, 2005

    2005

  21. [29]

    Geodesics in first passage percolation

    Christopher Hoffman. Geodesics in first passage percolation. Ann. Appl. Probab. , 18(5):1944--1969, 2008

    1944

  22. [30]

    Sublinear morse geodesics and first passage percolation, 2025

    Sagnik Jana and Yulan Qing. Sublinear morse geodesics and first passage percolation, 2025. https://arxiv.org/abs/2507.07859

    2025 arXiv

  23. [31]

    Geometry of geodesics through B usemann measures in directed last-passage percolation

    Christopher Janjigian, Firas Rassoul-Agha, and Timo Sepp\" a l\" a inen. Geometry of geodesics through B usemann measures in directed last-passage percolation. J. Eur. Math. Soc. (JEMS) , 25(7):2573--2639, 2023

    2023

  24. [32]

    Commensurated hyperbolic subgroups

    Nir Lazarovich, Alex Margolis, and Mahan Mj. Commensurated hyperbolic subgroups. Trans. Amer. Math. Soc. , 377(10):7377--7402., 2024

    2024

  25. [33]

    M. Mitra. Cannon- T hurston M aps for H yperbolic G roup E xtensions. Topology 37 , pages 527--538, 1998

    1998

  26. [34]

    M. Mitra. Cannon- T hurston M aps for T rees of H yperbolic M etric S paces. Jour. Diff. Geom.48 , pages 135--164, 1998

    1998

  27. [35]

    Cannon- T hurston maps

    Mahan Mj. Cannon- T hurston maps. In Proceedings of the I nternational C ongress of M athematicians--- R io de J aneiro 2018. V ol. II . I nvited lectures , pages 885--917. World Sci. Publ., Hackensack, NJ, 2018

    2018

  28. [36]

    Mj and P

    M. Mj and P. Sardar. A combination theorem for metric bundles . Geom. Funct. Anal. 22, no. 6 , pages 1636--1707, 2012

    2012

  29. [37]

    Un groupe hyperbolique est d\' e termin\' e par son bord

    Fr\' e d\' e ric Paulin. Un groupe hyperbolique est d\' e termin\' e par son bord. J. London Math. Soc. (2) , 54(1):50--74, 1996

    1996

  30. [38]

    Shalom and G

    Y. Shalom and G. Willis. Commensurated subgroups of arithmetic groups, totally disconnected groups and adelic rigidity. Geom. Funct. Anal. , 23(5):1631--1683, 2013

    2013

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