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Sublinear Morse Geodesics and First Passage Percolation

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A sublinearly Morse geodesic line guarantees a bi-infinite geodesic after random edge percolation.

desk verdict The claimed generalization of BT17 is plausible and the middle-recurrence section is a useful new tool, but the proof of the main theorem has a load-bearing gap in the upper-bound estimate that needs to be fixed before the result is established. read the letter →

arxiv 2507.07859 v1 pith:KKGL3HX4 submitted 2025-07-10 math.GT math.PR

Sagnik Jana , Yulan Qing This is my paper · ORCID
classification math.GTmath.PR MSC 60K3520F65
keywords firstpassagepercolationsublinearlyMorsegeodesicbi-infinitemiddlerecurrencesuperlineardivergencerandomedgelengthsbounded-degreegraph
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 shows that a geometric condition much weaker than hyperbolicity still forces first passage percolation to admit long geodesics. The condition is the existence of a sublinearly Morse bi-infinite quasi-geodesic line: a line whose quasi-geodesic tracking property weakens sublinearly as one moves away from the basepoint. If such a line exists in an infinite bounded-degree graph and the random edge lengths are positive, have finite mean, and have no atom at zero, then almost surely the random metric contains a bi-infinite geodesic line. The proof works by proving a middle recurrence property for sublinearly Morse lines and then a superlinear divergence estimate, which combine with standard first passage percolation bounds to rule out geodesics escaping to infinity.

What carries the argument

The load-bearing object is the middle-recurrence property of sublinearly Morse geodesic lines. A line is middle recurrent if any path whose endpoints lie on the line and whose length is at most $C$ times the endpoint distance must pass through a sublinear neighborhood of the middle third of the segment of the line between its endpoints. Proposition 3.5 proves that sublinearly Morse lines have this property, and Lemma 3.7 converts it into a superlinear lower bound on the length of any quasi-geodesic that runs outside a growing linear neighborhood of the line. That divergence estimate is what contradicts the almost sure linear upper and lower bounds for passage times of long paths, forcing the random geodesics to stay in a bounded set.

What would settle it

Find an infinite bounded-degree graph with a sublinearly Morse bi-infinite geodesic line and a sequence of paths whose endpoints lie on the line, whose length is at most $C$ times the endpoint distance, and which stay outside every sublinear neighborhood of the middle third of the line segment; this would refute Proposition 3.5 and remove the engine of Theorem A's proof. A finite computation on a candidate graph could search for such paths.

Watch

Extended reading notes

Core claim

The central claim is Theorem A: for an infinite connected bounded-degree graph $X$ and i.i.d. edge weights with $\mathbb{E}\omega_e < \infty$ and $\nu(\{0\})=0$, if $X$ contains a sublinearly Morse bi-infinite quasi-geodesic line, then for almost every $\omega$ there is a bi-infinite geodesic line in $X_\omega$. The proof actually produces the line within a finite distance of the basepoint, with the distance depending on the environment. This is established by taking $\omega$-geodesics between pairs of points on the initial line at distance $n$ on either side of the basepoint, showing these random geodesics cannot escape to infinity because their $\omega$-length would then have to grow both at least as fast as a superlinear function of $n$ and at most linearly in $n$; the contradiction forces them to accumulate, and Arzelà–Ascoli yields the limiting bi-infinite geodesic.

Load-bearing premise

The proof of the middle-recurrence proposition assumes, without proof, that in the contradiction setup the distances from the basepoint to the two endpoints of each escaping path grow at comparable linear rates; if that fails, the estimates that produce the contradiction no longer follow.

Editorial extensions

If this is right

  • If the theorem is correct, any infinite bounded-degree graph with a sublinearly Morse bi-infinite line has an almost-sure bi-infinite geodesic in every i.i.d. positive weight environment with finite mean and no zero atom.
  • The bi-infinite geodesic can be chosen to lie within a bounded distance of the original basepoint, so it is produced by a compactness limit rather than by following the original line.
  • This extends the earlier result for Morse quasi-geodesic lines to the strictly larger class of sublinearly Morse lines, for which the tracking neighborhood may grow sublinearly rather than remain bounded.
  • The middle-recurrence lemma gives a new characterization of sublinearly Morse lines that is available for first passage percolation arguments.

Reading between the lines

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

  • The authors leave open whether the exhibited bi-infinite geodesic can itself be chosen sublinearly Morse in the random metric; that preservation is not established here.
  • If the middle-recurrence and divergence pair is the real mechanism, then other geometric conditions that imply the same pair would also yield bi-infinite geodesics under first passage percolation, so the conclusion may extend beyond sublinearly Morse lines.
  • One could test the theorem's sharpness by looking for a graph with no sublinearly Morse bi-infinite geodesic but where first passage percolation still almost surely has one; the paper gives no evidence either way.
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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. The paper studies first passage percolation on infinite bounded-degree graphs. Theorem A asserts that if the underlying graph contains a sublinearly Morse bi-infinite quasi-geodesic line, then almost surely the random weighted graph admits a bi-infinite geodesic, and moreover this geodesic lies at uniformly bounded random distance from the basepoint. The proof follows the strategy of Benjamini--Tessera: establish a 'middle recurrence' property for sublinearly Morse lines (Proposition 3.5), derive a superlinear divergence estimate for paths leaving a neighborhood of the line (Lemma 3.7), and then take limits of finite geodesic segments between pairs of points on the line (Theorem 3.8).

Significance. If the proof can be completed, the result is a natural and valuable generalization of Benjamini--Tessera's theorem from Morse geodesics to sublinearly Morse geodesics, and it would further demonstrate the usefulness of the sublinear Morse boundary framework. The paper is clearly organized and engages with recent literature. However, as written, several load-bearing steps are not justified: Proposition 3.5 relies on unproved endpoint-growth assumptions and an unstated extension of a lemma of Aougab--Durham--Taylor, and the upper bound in Theorem 3.8 uses a uniform linear estimate for random edge weights on an infinite family of varying paths that does not follow from Proposition 2.13. These gaps are substantive rather than cosmetic.

major comments (4)
  1. [§3, Proposition 3.5] The proof asserts two 'without loss of generality' endpoint-growth assumptions: that eventually d(e_i,o) ≥ c3 d(s_i,o), and later that d(s_i,e_i) ≥ c2 d(o,s_i). These are not consequences of the failure of middle recurrence as stated; they are structural conditions on the specific paths constructed from the contradiction hypothesis. The estimates in (4) and (5), the bound on sl(p'_i), and the later claim that d(s'_i,e'_i) grows linearly with K_i all depend on these assumptions. Without a proof that one may arrange these conditions, Proposition 3.5 is not established.
  2. [§3, Proposition 3.5] The proof invokes 'a slight adaptation of Lemma 3.4' of ADT17 but does not state the adapted lemma or verify its hypotheses in the sublinear setting. In particular, the original lemma concerns a path at distance at least K from a contracting geodesic whose endpoints are exactly at distance K; here the path p'_i is outside a c1-linear neighborhood and the relevant distance K_i = d(s'_i, β) is not shown to have the properties required by the adapted statement. This needs a precise formulation and proof, not a citation to an adaptation.
  3. [§3, Lemma 3.7] The proof contains an internal inconsistency in the choice of R. It states 'R is an integer greater than d(x,y)', but the displayed inequality in the same paragraph, 4R ≤ d(x',y') ≤ 2R + d(x,y), implies d(x,y) ≥ 2R. The subsequent bound |p| ≤ 2R + |γ| ≤ d(x,y) + C d(x,y) also uses 2R ≤ d(x,y), which contradicts the stated choice of R. This step must be repaired; if R is instead constrained by R ≤ d(x,y)/2, the later assertion that one can choose d(x,y) large enough so that R ≥ c·κ'(d(o,p_i)) needs a separate justification, since R and d(x,y) are linked by the construction.
  4. [§4, Theorem 3.8] The upper bound in the proof, dω(γ_n^ω(p), γ_n^ω(q)) ≤ 2bR_n + 2r0 + |γ0([i,j])|ω, is justified by 'Line 6 and Proposition 2.13'. Line 6 only records the graph distances from γ_n^ω(p) and γ_n^ω(q) to fixed points on γ0; the actual paths realizing these distances are not fixed and vary with n. Proposition 2.13(1) gives a constant r0 for each fixed self-avoiding path, but it does not give a uniform r0 over the infinite family of length-R_n spokes. With i.i.d. finite-mean but possibly unbounded edge weights, the ω-length of such spokes is not uniformly bounded by a linear function of R_n plus a constant. Without this uniform estimate, the inequality (7) does not follow, and the lower bound alone is compatible with R_n = o(n) and slowly growing φ. This is a load-bearing gap in the contradiction argument.
minor comments (5)
  1. [Abstract and §1] The abstract says the conclusion is a bi-infinite geodesic line, but Theorem A states a 'bi-infinite geodesic ray'; these terms should be aligned, since the proof actually produces a bi-infinite line.
  2. [§2.3, Proposition 2.13] In part (1), the quantifier 'for all i ≤ 0 ≤ j' is unusual; it should be clarified whether the intended statement is for all i ≤ j with the segment [i,j] containing 0, or for all pairs (i,j) with i ≤ 0 ≤ j.
  3. [§3, Lemma 3.9] The notation γ0((∞,i]) should be γ0((-∞,i]); as written it is confusing.
  4. [Throughout] The basepoint is denoted o but in several inequalities, for example in Lemma 3.9, the text writes d(x,0) instead of d(x,o); this should be fixed for consistency.
  5. [§4, Theorem 3.8] The invocation of the Arzelà--Ascoli theorem on a countable graph is nonstandard; a diagonal extraction or Tychonoff compactness argument would be more precise, although the intended compactness statement is clear.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the FPP bi-infinite geodesic conclusion is not an input to any definition or prior result.

full rationale

The derivation chain does not reduce to its inputs. Theorem A's conclusion — existence of an almost-sure bi-infinite ω-geodesic — is not assumed in the hypotheses or in the cited sublinearly Morse framework. The paper uses [BT17, Lemma 2.3, 2.5] for fixed-path linear weight bounds and [QRT23, Prop A.6, Lemma 4.2] for contracting/geodesic properties; these are prior results with stated assumptions not containing the FPP conclusion, and the fact that QRT23 shares an author does not make the support circular. The upcoming self-citation [JQ] is only mentioned as future work, not load-bearing. The apparent issues in the proof — the unconditional 'without loss of generality' scaling assumptions in Proposition 3.5 and the application of Proposition 2.13(1) to the varying R_n-spokes in Theorem 3.8 — are correctness gaps, not circular reductions: they do not turn the conclusion into the hypothesis by construction. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported to force the argument. Hence no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters or empirical constants. The paper is a pure mathematical theorem and introduces no entities beyond the existing notion of sublinearly Morse lines and the new property 'middle recurrence', which is not a postulated entity with independent evidence.

assumptions (5)
  • domain assumption X is an infinite connected graph of bounded degree; edge weights are i.i.d. with ν({0})=0 and finite expectation.
    Hypotheses of Theorem A, used for Proposition 2.13 and to ensure geodesics exist.
  • domain assumption X contains a sublinearly Morse bi-infinite quasi-geodesic line.
    This is the input hypothesis; the result is conditional on it.
  • standard math Results of QRT23 (Prop 2.8 and Lemma 4.2(1)) are correct: κ-Morse implies κ′-weak contracting, and a sublinearly Morse quasi-geodesic line lies in a κ-neighborhood of a κ-Morse geodesic line.
    Imported from the authors' prior work; the proof starts by replacing the quasi-geodesic with a geodesic line.
  • ad hoc to paper Lemma 3.4 of ADT17 extends to sublinearly contracting sets.
    Used without proof in Proposition 3.5 to derive the bound on κ′(K_i)/K_i; this is a load-bearing unproved step.
  • ad hoc to paper The WLOG endpoint-growth assumptions in Proposition 3.5 (d(e_i,o) ≥ c3 d(s_i,o) and d(s_i,e_i) ≥ c2 d(o,s_i)) hold for the constructed paths.
    Stated without justification; if false, the estimates leading to the contradiction fail.

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Cite this review

Pith. "Pith review of Sublinear Morse Geodesics and First Passage Percolation." pith.science (2026). https://pith.science/paper/KKGL3HX4

@misc{pith2026250707859,
  author       = {Pith},
  title        = {Pith review of: Sublinear Morse Geodesics and First Passage Percolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KKGL3HX4}},
  note         = {Machine review of arXiv:2507.07859}
}
read the original abstract

Given an infinite connected graph, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges of the graph. Assume that the graph is infinite and of bounded degree. Assume also strict positivity and finite expectation of the edge length distribution and existence of a sublinearly Morse bi-infinite geodesic line, we prove that almost surely there exists a bi-infinite geodesic line. This generalizes a previous result of \cite{BT17} regarding Morse geodesics.

Figures

Figures reproduced from arXiv: 2507.07859 by the authors.

Figure 1
Figure 1. A κ-neighbourhood of a geodesic ray b with multiplicative constant n. Definition 2.4 (κ-Morse I, κ-Morse II). Let Z ⊆ X be a closed set, and let κ be a concave sublinear function. We say that Z is κ-Morse if one of the following equivalent (see Proposition 3.10 [QRT23]) condition holds: I. There exists a proper function mZ : R 2 → R such that for any sublinear function κ ′ and for any r > 0, there exists R such that… view at source ↗
Figure 2
Figure 2. Behavior of the path outside c1-linear neighborhood of κ-Morse geodesic line β. Since β is κ ′ -contracting with respect to distance from the base point. For any ball B disjoint from β and centered at a point x ∈ p ′ i , (3) diam(πβ(B)) < c2κ ′ (d(x, o)). Since x ∈ p ′ i is outside of the c1-linear neighborhood, we have that c1d(x, o) ≤ d(x, β) thus we have diam(πβ(B)) < c2κ ′ (d(x, o)) ≤ c2κ ′ ( 1 c1 d(x, β)) ≤ c2 … view at source ↗
Figure 3
Figure 3. Path γn joining xn, yn are outside the ball B(0, 10Rn) vertex x ∈ γ0(−∞, i), 10Rn ≤ d(γ n ω (r), y) + d(y, x) = d(γ n ω (r), x) Therefore, d (γ n ω (r), γ0((∞, i])) ≥ 10Rn. and since we also have d (γ n ω (r), o) ≥ 10Rn, we deduce that d (γ n ω (r), γ0) ≥ 5Rn If P = max{k ≤ r : d(γ n ω (k), γ0) ≤ Rn} and Q = min{k ≥ r : d(γ n ω (k), γ0) ≤ Rn} then we get p ≤ P and q ≥ Q be the largest and the smallest integer respec… view at source ↗

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