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On the most likely geodesic in last passage percolation

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The most likely geodesic in last passage percolation is asymptotically the corner path that runs along the edges of the cube.

desk verdict A solid, honestly-flagged conditional result: the sharp √n coefficient rests on an unproved moderate-deviation range, but the unconditional theorems are real and worth serious refereeing. read the letter →

arxiv 2608.11378 v1 pith:UE3CR5OU submitted 2026-08-11 math.PR

Sam McKeown , Shuta Nakajima This is my paper · ORCID
classification math.PR MSC 60K3560F1082B43
keywords lastpassagepercolationgeodesicscornerpathmoderatedeviationsexponentialweightsBernoullimonotonicityorderinglarge
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

The paper asks which path is most likely to be the geodesic in last passage percolation, and argues that the answer is an extreme corner path that runs straight along the edges of the cube. Under a sharp but not-yet-proved moderate deviation bound, it shows in two-dimensional exponential last passage percolation that the corner path's log-probability is $-2nI(\mu)+\frac{4}{3}\sqrt{2n}+o(\sqrt{n})$, and that this equals the maximum over all paths up to the $\sqrt{n}$ order, making the corner path nearly modal. Unconditionally, it proves the corner path beats every path with linearly many corners by an exponential factor in any subcritical tiltable distribution, and for small Bernoulli weights it proves a monotonicity ordering under which the corner path is uniquely most likely. A sympathetic reader cares because these results turn a heuristic about entropic competition into explicit asymptotic formulae for the probability that a given path is the geodesic.

What carries the argument

The central object is the corner path $\gamma^\ulcorner$, which for the cube $[0,n]^d$ is the path with the minimal number of corners ($d-1$). The quantitative results are carried by an exponent-balancing argument: the probability that a fixed path is the geodesic is bounded above by a sum over thresholds $a$ of $\mathbb{P}(L(\gamma)\ge a)\mathbb{P}(L_n\le a+1)$, by negative correlation of the two monotone events. The first factor is the large-deviation tail of the path's own weight, $e^{-2nI(\mu)+\frac12 t n^{1/3}}$, and the second is the passage-time lower tail, $e^{-\frac{1}{192}t^3}$, from the moderate deviation estimate (3). Maximizing $\frac12 t n^{1/3} - \frac{1}{192}t^3$ gives $t_* = 4\sqrt{2}\,n^{1/6}$ and the value $\frac43\sqrt{2n}$ that appears in the theorem. For the matching lower bound, the proof constructs an event in which every bulk shortcut is suppressed: the corner weights beat all point-to-line passage times (Lemma 3), and the point-to-line lower-tail bound is upgraded to a maximum over nearby starting points (Lemma 4), allowing the weights in the two endpoint segments of length $\asymp n^{2/3}$ to be relaxed by an $O(n^{1/3})$ amount. The sharp point-to-line bound of Assumption 3 is exactly what makes this relaxation cost match the upper bound; without it, only the weaker constant $1/(3\sqrt{3c_l})$ is obtained.

What would settle it

Compute the point-to-line lower-tail constant in exponential LPP in the regime $t \asymp n^{1/6}$: if $\log P(L(0,0;n)\le 2\mu n - t n^{1/3})/t^3$ does not tend to $-1/48$, Assumption 3 fails and the matching limits in Theorem 2 are not the true asymptotics. Alternatively, for Bernoulli weights with $p\le p_n$ (the explicit polynomial threshold in the proof of Theorem 3), enumerate all paths for small $n$ and check that every bump or unwind move strictly increases the geodesic probability; a counterexample refutes the monotonicity claim.

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Extended reading notes

Core claim

In two-dimensional exponential last passage percolation, the paper's central claim is that the corner path $\gamma^\ulcorner$, which goes $(0,0)\to(0,n)\to(n,n)$, has probability of being the unique geodesic equal to $$\log P(\gamma^\ulcorner = \gamma^*_n) = -2n I(\mu) + \frac{4}{3}\sqrt{2n} + o(\sqrt{n}),$$ and that the same expression holds for $\max_{\gamma\in\Gamma_n}\log P(\gamma = \gamma^*_n)$, so the corner path attains the maximum probability up to $o(\sqrt{n})$ corrections. This is established conditionally on Assumption 3, a uniform sharp lower-tail moderate deviation bound for point-to-line passage times that is not currently proved for the needed range; the unconditional results give a limsup of $4/3$ for the normalized maximum and a liminf of $1/(3\sqrt{3c_l})$ for the corner path, where $c_l>0$ is the point-to-line lower-tail constant. The paper also claims that for general tiltable subcritical weights in $\mathbb{Z}^d$, any path with at least $\epsilon n$ corners has probability at most $C_\epsilon e^{-n/C_\epsilon}$ times that of the corner path, and that for Bernoulli weights with sufficiently small $p$ in $d=2$, the two corner paths are exactly the unique most likely geodesics, with a monotonicity ordering (bump and unwind moves) that makes probabilities increase as a path is smoothed toward the corners.

Load-bearing premise

The load-bearing premise is Assumption 3: the point-to-line lower-tail moderate deviation bound $\log P(L(0,0;n)\le 2\mu n - t n^{1/3}) \ge -\frac{1}{48}t^3 + o(t^3)$ must hold uniformly for $T\le t\le \rho n^{1/6+\zeta}$, a range not covered by the best available estimates (which reach only $t=O(n^{1/10})$); if this bound is false or has a different constant, the matching limits in Theorem 2 and expression (5) collapse.

Editorial extensions

If this is right

  • In two-dimensional exponential LPP, the probability that the corner path is the geodesic is asymptotically $\exp(-2n(1-\log 2) + \frac{4}{3}\sqrt{2n} + o(\sqrt{n}))$, and the most likely geodesic has the same asymptotic probability up to $o(\sqrt{n})$; the two corner paths are equally likely and dominate every other path at this order.
  • The set of paths whose geodesic probability is at least that of the corner path has size at most $e^{\epsilon n}$ under the subcritical tiltability assumptions, whereas the total number of directed paths grows like $d^{dn}$; hence almost all paths are exponentially less likely than the corner path.
  • For Bernoulli weights with $p \le p_n$, the corner path and its reflection are the unique most likely geodesics, and the partial order generated by bump and unwind moves is monotone: applying these moves can only increase the probability of being the geodesic.
  • Unconditionally, the normalized maximum log-probability satisfies $\limsup_{n} (\max_\gamma \log P(\gamma=\gamma^*_n)+2nI(\mu))/\sqrt{2n} \le 4/3$, so the constant $4/3$ is an upper barrier that any sharper modal result must meet.

Reading between the lines

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

  • The balancing exponent $t_* = 4\sqrt{2}\,n^{1/6}$ should be universal for the $\sqrt{n}$ correction in exactly solvable LPP models; replacing the rate function $I$ and the moderate deviation constant $1/192$ by their geometric or Brownian counterparts yields the corresponding modal constant.
  • Lemma 4's method of upgrading a pointwise lower-tail bound to a maximum over a small number of correlated passage times at a super-exponential penalty is a transferable technique for problems in random growth models where many dependent passage times must be controlled simultaneously.
  • The monotonicity ordering for Bernoulli weights is proved only for polynomially small $p$; the signed-area formula for $C^{1,1}$ suggests the higher configuration counts $C^{a,b}$ with $a+b \ge 3$ are the only obstruction, so a natural testable conjecture is that the same ordering holds for every subcritical $p<1/2$.
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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

3 major / 5 minor

Summary. The paper studies the probability that a prescribed path is the geodesic in last passage percolation on Z^d, with emphasis on the planar exponential case. The authors introduce three mechanisms favoring 'corner' paths and prove: (i) unconditionally, under subcritical Assumptions 1 and 2, any path with linearly many corners is exponentially less likely to be the geodesic than the corner path (Theorem 1 and Corollary 1); (ii) in exponential LPP, an unconditional upper bound limsup ≤ 4/3 for the √n correction to the modal probability, and a lower bound of 1/(3√(3c_l)) in terms of an unspecified constant c_l, with equality to 4/3 under an explicit Assumption 3 on the sharp lower-tail moderate deviation rate for point-to-line times (Theorem 2); (iii) a conditional comparison showing corner paths beat staircase paths with k≥8 segments (Corollary 2); and (iv) a monotonicity theorem for Bernoulli weights with small p under bump and unwind moves (Theorem 3). The paper is transparent about the unproved nature of Assumption 3.

Significance. If the sharp results were unconditional, the paper would make a substantial contribution: it gives the first asymptotically sharp two-term expansion for the modal geodesic probability in exponential LPP, with a parameter-free constant 4/3 arising from optimizing competing exponents, and it rigorously establishes a corner-path optimality principle in several settings. The unconditional Theorem 1 and Corollary 1 are solid and of independent interest, and Theorem 3 provides a complete small-p monotonicity picture. The honest flagging of Assumption 3 is a strength, but it means the headline sharp equality is conditional on an open moderate-deviation estimate. The paper does not rely on self-citation or fitted parameters; the constant comes from the structure of the moderate deviation exponents.

major comments (3)
  1. [Section 2, Theorem 2 and Assumption 3] The sharp equality in Theorem 2, the displayed expression (5), and the claim that corner paths are modal up to √n order all rest on Assumption 3, which requires the log-probability lower bound −(1/48)t^3 + o(t^3) uniformly for T ≤ t ≤ ρ n^{1/6+ζ}. As the paper notes, the best available result [Bas+25, Theorem 1.6] covers only t = O(n^{1/10}), so the required range is not proved. Without Assumption 3, the unconditional lower bound in Theorem 2 is only liminf ≥ 1/(3√(3c_l)) with c_l an unspecified constant from [Bas+21, Theorem 2]. The authors should either prove the needed moderate-deviation range or explicitly present Theorem 2 and (5) as conditional on an open conjecture and discuss the status of that conjecture in more detail.
  2. [Section 4, Lemma 4] The proof of Lemma 4 transfers the diagonal point-to-line moderate deviation lower bound to the maximum over starting points (0,k) with k ≤ a = ⌊n^{2/3−ε}⌋ by using a lower-tail penalty for the non-diagonal point-to-point passage times X_k over distance b = n^{1−ε}. The text states that one should prove uniformity across directions and that the bound from [LR10, Theorem 2] suffices, but no proof or precise statement of the required non-diagonal bound is supplied. Since Lemma 4 is used in the lower-bound part of Theorem 2 and in the derivation of the 4/3 constant under Assumption 3, this gap must be closed or explicitly referenced.
  3. [Section 4, Corollary 2] The proof of Corollary 2 contains two explicit omissions: the off-maximum terms in the union bound are not bounded ('We do not show carefully how to bound the off-maximum terms for reasons of length'), and the argument assumes a sharp half-space moderate deviation constant c_h = 1/384 over a wide but unspecified range of t. As stated, Corollary 2 is therefore not proved. The corollary should be restated as conditional on the same assumptions used in its proof, or the missing estimates must be supplied.
minor comments (5)
  1. [Section 4, proof of the upper bound in Theorem 2] The uniform moderate deviation estimate (3) is quoted for T ≤ t ≤ ρ n^{2/3}, while the proof uses it only for t ≤ ρ n^{1/2}; the reference to [Led18] is described as 'an indication of the proof', so a precise statement of the proven range and a complete citation for exponential weights would improve the paper.
  2. [Section 4, Corollary 2 statement] The statement says 'take k ≥ K, where K is a deterministic constant' but the value K is not specified; the proof indicates k ≥ 8 under the auxiliary assumptions on c_l and c_h. The statement should either specify the assumptions or the constant.
  3. [Section 1, Figure 2 caption] The caption says the right panel is for n = 30, but the displayed matrix appears to be 8×8; please check the figure and caption.
  4. [Section 3, proof of Theorem 1] The notation p_0 in the displayed bound is used both as the limit of p_α as α→0 and as a fixed constant p_0 > 0 in Lemma 3 of Section 4; this clash of notation could confuse readers and should be resolved.
  5. [Appendix C, Proposition 3] The proof says 'The case α = 1 has been treated in Appendix B', but Appendix B concerns the corner-path probability, not the transversal fluctuation bound; please add the missing reference or explanation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the 4/3 coefficient arises from optimizing external moderate-deviation exponents, and the unproved range in Assumption 3 is an openly stated condition, not a fitted or self-referential input.

full rationale

The derivation chain is self-contained relative to external moderate-deviation inputs, and no step reduces to its own conclusion. In Theorem 2, the upper bound is unconditional: expression (3), an external lower-tail moderate deviation estimate for point-to-point passage times with constant 1/192, is combined with the Bahadur-Rao tail (2) for a fixed path; the coefficient 4/3 emerges from the explicit optimization max_t [(1/2)t n^{1/3} - (1/192)t^3] at t = 4 sqrt(2) n^{1/6}, not from matching the claimed answer. The lower bound uses Assumption 3, which explicitly sets c_l = 1/48 in (4); the displayed optimization max_alpha [alpha/2 - c_l alpha^3] gives 1/(3 sqrt(6 c_l)) per side, and with c_l = 1/48 this equals the same 4/3. Thus the sharp equality in Theorem 2 is conditional on an external estimate, but the constants are not fitted to the target probability and no quantity is defined in terms of the asserted conclusion. The paper honestly flags the limiting gap: it notes that the best available result [Bas+25, Theorem 1.6] covers only t = O(n^{1/10}) while Assumption 3 requires t = O(n^{1/6}), and it records that the sharp half-space range needed for Corollary 2 is also assumed; these are correctness risks, not circularity. There are no load-bearing self-citations by the present authors, and the uniqueness of geodesics is standard rather than imported from prior work. Theorem 1 rests on independent large-deviation bounds [CZ03, Alb+25] plus a self-contained Appendix B, and Theorem 3 is a combinatorial counting argument. Consequently the paper is a conditional derivation from stated external assumptions, with no circular reduction found.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central result relies on standard large deviation machinery (Cramér, Bahadur-Rao, Harris, Kingman) and on several model-specific assumptions. The most load-bearing are Assumptions 1 and 2 for the general theorem, and Assumption 3 plus the half-space constant assumption for the sharp conditional statements. No parameters are fitted to data; the constants 4/3 and 1/48 come from external moderate deviation results.

assumptions (6)
  • domain assumption Assumption 1: P(ω_x < -t) < e^{-c t^ν} for some ν>1, c>0
    Controls the lower tail of the weight distribution; used in Appendix A to prove the lower-tail large deviation bound P(L_n < dn(µ-ε)) < e^{-An}.
  • domain assumption Assumption 2: the time constant µ is tiltable, µ < sup_{t≥0} Λ'(t)
    Ensures an exponential tilt can raise the mean above µ; used throughout Theorem 1 and Appendix B. Explicitly rules out critical/supercritical regimes.
  • ad hoc to paper Assumption 3: log P(L(0,0;n) ≤ 2µn - t n^{1/3}) ≥ -(1/48)t^3 + o(t^3) uniformly for T ≤ t ≤ ρ n^{1/6+ζ}
    Stated as an explicit assumption; the known result [Bas+25, Theorem 1.6] gives only t=O(n^{1/10}), not the needed t=O(n^{1/6}). Required for the sharp constant 4/3 in Theorem 2.
  • ad hoc to paper Sharp half-space moderate deviation constant c_h = c_f/2 = 1/384 over a wide range
    Used in Corollary 2 to compare staircase paths; the paper says this follows from [Bas+25] only if it holds across a wide enough range, which is assumed.
  • ad hoc to paper Non-diagonal uniformity of the moderate deviation bound (3), or a cruder bound from [LR10, Theorem 2]
    Lemma 4 uses (3) for point-to-line times in directions other than the diagonal; the paper flags that (3) is proved only for the diagonal and states a cruder bound suffices, without supplying the argument.
  • domain assumption Tracy-Widom GOE convergence for point-to-line passage times and coalescence estimates for maxima over intervals of length ≪ b^{2/3}
    Used in Lemma 3 to get P(M_{a,b} ≤ 0) ≥ p > 0 for short intervals; cited from [BR01], [Bis18], [Zha20].

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Pith. "Pith review of On the most likely geodesic in last passage percolation." pith.science (2026). https://pith.science/paper/UE3CR5OU

@misc{pith2026260811378,
  author       = {Pith},
  title        = {Pith review of: On the most likely geodesic in last passage percolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UE3CR5OU}},
  note         = {Machine review of arXiv:2608.11378}
}
read the original abstract

We consider the problem of identifying the paths most likely to occur as geodesics in last passage percolation. Heuristics suggest that these modal paths should be the extreme corner paths, going straight between the corners of the cube of accessible vertices. We identify three mechanisms which favour such corner paths, and show that they are more likely to appear than all but a vanishingly small portion of paths. We make more specific comparisons in exponential last passage percolation, where moderate deviation estimates may be used to show that corner paths are nearly modal in a precise sense. Finally, we show a form of monotonicity of geodesic probabilities in a special case and conjecture that this holds in general.

Figures

Figures reproduced from arXiv: 2608.11378 by the authors.

Figure 1
Figure 1. The observed frequency of individual geodesics when [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The observed probabilities of a site being visited by a geodesic, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Bumping a corner of a path. The concavity formed by [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Unwinding a path about two corners. Observe that the order of [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: The staircase γ k n , shown with k = 3. We consider a half-space geodesic lying below the staircase and many geodesics going through the bulk of the staircase’s steps. The cyan paths are unrestricted point-to-point geodesics, while the outermost magenta paths are half-…
Figure 6
Figure 6. Figure 6: Some of the notation introduced for points on the staircase [PITH_FULL_IMAGE:figures/full_fig_p029_6.png]
Figure 7
Figure 7. Figure 7: An illustration of the signed area D(γ, π) between two paths. The dotted lines and holes denote vertices on the boundary which are not in￾cluded. The interior vertices and remaining boundary vertices are included. Proof. Write γ = (γ 0 , . . . , γ2n ) and γ i = (γ i 1 …

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