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Last passage percolation in hierarchical environments

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Pith's one-line read For i.i.d. weights with P(X>t)~Ct^{-2}, the last passage time is shown to be at least c n(log n)^{3/4}/log log n with high probability, and a finite-second-moment distribution is exhibited whose last passage time grows superlinearly.

arxiv 2411.08018 v3 pith:3OKUBTCG submitted 2024-11-12 math.PR math-phmath.MP

Shirshendu Ganguly , Victor Ginsburg , Kyeongsik Nam This is my paper · ORCID
classification math.PRmath-phmath.MP
keywords environmentscriticalhierarchicalrandomdirectedenvironmentexponentsanalysis
verification ladder T0 review T1 audit T2 compute T3 formal

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The reading

The paper studies last passage percolation, a model where a directed path moves from corner to corner of a large square, collecting random weights along the way, and the quantity of interest is the maximum total weight. For ordinary light-tailed random weights, fluctuations are described by the KPZ universality class. This paper looks at two critical environments where the usual theory is not expected to apply. The first is independent weights with a power-law tail of exponent 2, the conjectured boundary between KPZ behavior and heavy-tailed behavior. The second is the branching random walk, a hierarchical stand-in for the two-dimensional Gaussian free field, where the field has logarithmic correlations. The authors develop a multi-scale construction that builds a high-weight path scale by scale. For the heavy-tailed model, they prove that with overwhelming probability the passage time is at least a constant times n(log n)^{3/4}/log log n, strictly larger than the linear order. They also prove concentration bounds. For the branching random walk, the lower bound is n(log n)^{1/2}/log log n, and the fluctuations are shown to be of order n. A byproduct is a negative answer to a 2002 question of Martin: a weight distribution with finite second moment can still have a superlinear passage time, so finite second moment is not sufficient for linear growth. The results are bounds, not exact exponents; the true growth rates remain open. The paper's contribution is a rigorous framework for these critical hierarchical environments, where even non-rigorous exponent predictions had been missing.
Extended reading notes

Core claim

Theorem 1: For i.i.d. nonnegative weights with P(X>t) ~ C t^{-2}, there exists c>0 such that for all large n, P(L_n >= c n (log n)^{3/4} / log log n) >= 1 - e^{-(log n)^97}. Theorem 5: there exists a distribution with finite second moment such that L_n/n -> infinity almost surely, answering Martin's question in the negative. If the paper is correct, the critical heavy-tailed LPP grows strictly faster than linearly with a polylogarithmic correction, and finite second moment does not imply a finite limit shape.

Load-bearing premise

The most fragile premise is the geometric slope control underlying the multi-scale construction. Lemma 4.3(ii) and equations (4.17)-(4.18) require that the slopes of all rectangles at all scales remain within a sub-polynomial factor of 1, roughly e^{O(sqrt(log n / log log n))}. If this failed, the cylinders Cyl_{rho/lambda^2}(R_{i,j}) would not be disjoint and Proposition 4.4 would not provide the required number of large weights. The assumptions that make this work are the exact inverse-square tail via (4.1) and the elementary inequalities in Lemmas 3.5 and 4.6; all are proved, but the entire logarithmic-correction exponent depends on their uniform validity.

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Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No parameter is fitted to data. The listed parameters are proof constants chosen by hand to balance slope loss against concentration; they are not model inputs. All external inputs are standard theorems or prior LPP results, none of which already contain the target lower bounds. No new particles, fields, or entities are postulated.

free parameters (3)
  • zeta = 100 log log n
    Scale separation in the multi-scale construction (Eq. (4.15)); chosen by hand for concentration. The theorem holds for any sufficiently large constant multiple of log log n.
  • lambda = (log n)^{1/4}
    Cylinder width parameter in the construction (Eq. (4.2)); chosen by optimizing the tradeoff in Section 2 and determines the exponent 3/4 in Theorem 1.
  • rho = (log log n)^{1/2}
    Cylinder widening factor (Eq. (4.4)); the proof only needs 1 << rho <= (log log n)^{1/2}.
assumptions (5)
  • standard math Efron-Stein inequality for variance of functions of independent variables
    Used in Theorems 2 and 4 via Proposition 5.1 and the variant in Section 6.1.
  • standard math Pick's theorem for lattice polygons
    Used in Lemma 3.2 to lower-bound the number of lattice points in cylinders.
  • domain assumption Known LPP upper bounds for Bernoulli and Poisson weights from [AD95] and [Mar02, Theorem 2.3]
    Used in Section 2 and Theorem 2 to bound expected passage times under truncated and squared weights.
  • domain assumption Maximum of the branching random walk is O(log n), from [Zei16]
    Used to justify the trivial O(n log n) upper bound for BRW LPP in Section 2.
  • domain assumption Hambly-Martin results for alpha in (0,2) and Martin's linear growth criteria from [HM07] and [Mar02]
    Used for context and comparison; the main construction does not rely on these results.

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Pith. "Pith review of Last passage percolation in hierarchical environments." pith.science (2026). https://pith.science/paper/3OKUBTCG

@misc{pith2026241108018,
  author       = {Pith},
  title        = {Pith review of: Last passage percolation in hierarchical environments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3OKUBTCG}},
  note         = {Machine review of arXiv:2411.08018}
}
read the original abstract

Last passage percolation (LPP) is a model of a directed metric and a zero-temperature polymer where the main observable is a directed path evolving in a random environment accruing as energy the sum of the random weights along itself. When the environment has light tails and a fast decay of correlation, the fluctuations of LPP are predicted to be explained by the Kardar-Parisi-Zhang (KPZ) universality theory. However, the KPZ theory is not expected to apply for many natural environments, particularly "critical" ones exhibiting a hierarchical structure often leading to logarithmic correlations. In this article, we initiate a novel study of LPP in such hierarchical environments by investigating two particularly interesting examples. The first is an i.i.d. environment but with a power-law distribution with an inverse quadratic tail decay which is conjectured to be the critical point for the validity of the KPZ scaling relation. The second is the Branching Random Walk which is a hierarchical approximation of the two-dimensional Gaussian Free Field. The second example may be viewed as a high-temperature (weak coupling) directed version of Liouville Quantum Gravity, which is a model of random geometry driven by the exponential of a logarithmically correlated field. Due to the underlying fractal structure, LPP in such environments is expected to exhibit logarithmic correction terms with novel critical exponents. While discussions about such critical models appear in the physics literature, precise predictions about exponents seem to be missing. Developing a framework based on multi-scale analysis, we obtain bounds on such exponents and prove almost optimal concentration results in all dimensions for both models. As a byproduct of our analysis we answer a long-standing question of Martin on necessary and sufficient conditions for the linear growth of the LPP energy in i.i.d. environments.

Figures

Figures reproduced from arXiv: 2411.08018 by the authors.

Figure 1
Figure 1. A simulation of the directed geodesic from (0, 0) to (215 , 2 15) in an environment of i.i.d. weights with power-law tails of exponent 2. The left figure depicts the geodesic (blue) with two of its “skeletons” superimposed on top. The k th skeleton is obtained from the geodesic by linearly interpolating between the weights that exceed 215−k . The right figure depicts the same situation as the left, but with more ske… view at source ↗
Figure 2
Figure 2. Simulations of the directed geodesic in an environment of i.i.d. weights with power-law tails of exponent α = 2. We now proceed to our main results. Our first result is an explicit lower bound for the logarithmic correction exponent for the last passage time in an environment of i.i.d. weights with power-law tails of exponent α = 2. A subsequent result states an analogous bound for the last passage time on the branc… view at source ↗
Figure 3
Figure 3. A simulation of the directed geodesic from (0, 0) to (213 , 2 13) on the branching random walk. Our argument for Theorem 1 with minor modifications also allows us to construct a distribution with finite second moment and superlinear last passage time, thereby answering a question of Martin [Mar02]. Before stating the formal result we briefly review the pertinent literature on last passage limit shapes. 1.1.2. A dist… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Depicted are [0, n] 2 (black square) and the cylinder (shaded gray) around the diagonal of width √ δ, divided into smaller cylinders of length √ 1 δ . cylinder leads to an overall passage time of order n √ δ = √ n log n collected from ω. (Note that in our application, …
Figure 5
Figure 5. Figure 5: The black rectangle is Rect(a, b). The shaded gray region is the cylinder Cylr (Rect(a, b)), for some r ∈ (0, 1). The following lemma asserts a lower bound for the number of lattice points within a cylinder in terms of the area of that cylinder. This will allow us to w…
Figure 6
Figure 6. Figure 6: An illustration of Lemma 3.5. The rectangles Rect(a, b), Rect(b, c) and Rect(c, d) have identical slopes, and Rect(b, c) has (weakly) larger side lengths than the other two rectangles. Shaded in gray are the cylinders Cylr (Rect(a, b)) and Cylr (Rect(c, d)). Among all …
Figure 7
Figure 7. Figure 7: The construction of V (ℓ+1) . Step 1: Depicted are the rectangles R (ℓ) i and R (ℓ) i+1. By definition, their bottom-left and top-right corners are consecutive vertices in V (ℓ) . The set V (ℓ+1) is constructed by adding vertices from each such rectangle to V (ℓ) accor…
Figure 8
Figure 8. Figure 8: The proof of Lemma 4.3. 4.3. Vertex sets are large. In this subsection we prove Proposition 4.2(iii). For the reader’s convenience we recall the parameters defined previously in (4.2), (4.4): ζ := 100 log log n, M :=  log n 10ζ  , λ := (log n) 1/4 , ρ := (log log n) …
Figure 9
Figure 9. Figure 9: Depicted is [0, n] 2 (large black square) partitioned into s × s boxes (smaller black squares). Since the geodesic Γ (red) is directed, there exist points b a, b ∈ Γ with b a ≺ b such that the rectangle R∗ := Rect(a, b) (green) has area at least s 2/4 and is contained …
Figure 10
Figure 10. Figure 10: The multi-scale construction underlying Theorem 3. Left: The large black square has side length n 2 ℓζ , and the smaller squares inside have side length n 2 (ℓ+1)ζ (as drawn, 2ζ = 4). The up-skeleton consists of the two black points together with the three blue points…

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Works this paper leans on

15 extracted references · 4 canonical work pages

  1. [1]

    Hammersley’s Interacting Particle Process and Longest In- creasing Subsequences

    [AD95] D. Aldous and P. Diaconis. “Hammersley’s Interacting Particle Process and Longest In- creasing Subsequences”. Probability Theory and Related Fields 103.2 (1995), 199–213. doi: 10.1007/BF01204214 (cited on page 11). [ADH17] Antonio Auffinger, Michael Damron, and Jack Hanson. 50 years of first-passage percolation . University Lecture Series Vol

    doi:10.1007/bf01204214 1995

  2. [6]

    First Passage Percolation on the Exponential of Two- Dimensional Branching Random Walk

    Sections 12–14 . EMS Press, Berlin, 2023, pp. 4212–4244. isbn: 978-3-9854756-4-3. doi: 10.4171/icm2022/40 (cited on page 4). [DG17] Jian Ding and Subhajit Goswami. “First Passage Percolation on the Exponential of Two- Dimensional Branching Random Walk”. Electronic Communications in Probability 22 (2017), 1–14. doi: 10.1214/17-ECP102 (cited on page 14). [D...

    arXiv 2017

  3. [7]

    Kardar-Parisi-Zhang Universality

    isbn: 978-3-319-50486-5. doi: 10.1007/978- 3- 319- 50487- 2(cited on page 3). [Cor16] Ivan Corwin. “Kardar-Parisi-Zhang Universality”. Notices of the American Mathematical So- ciety 63.03 (2016), 230–239. doi: 10.1090/noti1334 (cited on page 3). [Cox89] H. S. M. Coxeter. Introduction to Geometry, Second Edition . Wiley Classics Library ed. Wiley,

    doi:10.1007/978- 2016

  4. [10]

    Directed polymers in a random environment with heavy tails

    1214/13-AOP858 (cited on page 9). [AL11] Antonio Auffinger and Oren Louidor. “Directed polymers in a random environment with heavy tails”. Communications on Pure and Applied Mathematics 64.2 (2011), 183–204. doi: 10.1002/cpa.20348 (cited on page 10). [BBP07] Giulio Biroli, Jean-Philippe Bouchaud, and Marc Potters. “Extreme Value Problems in Ran- dom Matri...

    work page doi:10.1002/cpa.20348 2011

  5. [12]

    Heavy Tails in Last-Passage Percolation

    doi: 10.1090/noti2059 (cited on page 4). [HM07] Ben Hambly and James B. Martin. “Heavy Tails in Last-Passage Percolation”. Probability Theory and Related Fields 137.1 (2007), 227–275. doi: 10.1007/s00440-006-0019-0 (cited on pages 4, 6, 10, 42). [HW65] J. M. Hammersley and D. J. A. Welsh. “First-Passage Percolation, Subadditive Processes, Stochastic Netwo...

    arXiv 2007

  6. [16]

    Growth Anomaly and Its Implications

    American Mathematical Society, 2016, pp. 437–471. isbn: 978-1-4704-2248-6 (cited on page 10). [Zha90] Yi-Cheng Zhang. “Growth Anomaly and Its Implications”. Physica A: Statistical Mechanics and its Applications 170.1 (1990), 1–13. doi: 10.1016/0378- 4371(90)90083- 5 (cited on page 3). [Zyg24] Nikos Zygouras. “Directed Polymers in a Random Environment: A R...

    arXiv 1990

  7. [91]

    American Mathematical Society, 2015, pp

    Proceedings of Symposia in Pure Mathematics. American Mathematical Society, 2015, pp. 155–214. isbn: 978-1-4704-2248-6. arXiv:

    work page 2015

  8. [1180]

    The Ergodic Theory of Subadditive Stochastic Processes

    Springer Berlin Heidelberg, 1986, pp. 125–264. isbn: 978-3-540-16441-8. doi: 10.1007/BFb0074919 (cited on page 9). [Kin68] J. F. C. Kingman. “The Ergodic Theory of Subadditive Stochastic Processes”. Journal of the Royal Statistical Society: Series B (Methodological) 30.3 (1968), 499–510. doi: 10.1111/j. 2517-6161.1968.tb00749.x (cited on pages 3, 9). [Kru...

    arXiv 1968

Show all 15 references
  1. [1212]

    Surface Growth with Power-Law Noise in 2+1 Dimensions

    3351 (cited on page 3). [BKW91] R. Bourbonnais, J. Kert´ esz, and D. E. Wolf. “Surface Growth with Power-Law Noise in 2+1 Dimensions”. Journal de Physique II 1.5 (1991), 493–500. doi: 10.1051/jp2:1991183 (cited on page 3). [BL21] Quentin Berger and Hubert Lacoin. “The scaling ...

    1991 doi

  2. [1983]

    Fractals in Surface Growth with Power-Law Noise

    isbn: 978-1- 4612-5451-5. doi: 10.1007/978-1-4612-5449-2 (cited on page 3). REFERENCES 47 [LS92a] Chi-Hang Lam and Leonard M. Sander. “Fractals in Surface Growth with Power-Law Noise”. Journal of Physics A: Mathematical and General 25.3 (1992), L135. doi: 10.1088/0305- 4470/25...

    1992 doi

  3. [1989]

    Liouville First-Passage Percolation: Subsequential Scaling Limits at High Temperature

    isbn: 978-0-471-50458-0 (cited on page 15). [DD19] Jian Ding and Alexander Dunlap. “Liouville First-Passage Percolation: Subsequential Scaling Limits at High Temperature”. The Annals of Probability 47.2 (2019), 690–742. doi: 10.1214/ 18-AOP1267 (cited on page 14). [DDDF20] Jia...

    2019 doi

  4. [1991]

    Linear Growth for Greedy Lattice Animals

    isbn: 978-3-642-20212-4. doi: 10.1007/978-3-642-20212-4 (cited on page 8). [Mar02] James B. Martin. “Linear Growth for Greedy Lattice Animals”. Stochastic Processes and their Applications 98.1 (2002), 43–66. doi: 10 . 1016 / S0304 - 4149(01 ) 00142 - 9(cited on pages 1, 8, 9, ...

    2002

  5. [2013]

    CLT for Spectra of Submatrices of Wigner Random Matrices

    isbn: 978-0-19- 953525-5. doi: 10.1093/acprof:oso/9780199535255.001.0001 (cited on pages 30, 40). [Bor10] Alexei Borodin. “CLT for Spectra of Submatrices of Wigner Random Matrices” (2010). arXiv: 1010.0898. Pre-published (cited on page 7). [BP24] Nathana¨ el Berestycki and Ell...

    2010

  6. [2017]

    Scaling of Surface Fluctuations and Dynamics of Surface Growth Models with Power-Law Noise

    161 pp. isbn: 978- 1-4704-4183-8. arXiv: 1511.03262 (cited on pages 3, 9). [AF91] J. G. Amar and F. Family. “Scaling of Surface Fluctuations and Dynamics of Surface Growth Models with Power-Law Noise”.Journal of Physics A: Mathematical and General 24.2 (1991), L79. doi: 10.108...

    1991 arXiv

  7. [2021]

    Departures from Many Queues in Series

    isbn: 978-1-108-84396-6. doi: 10 . 1017 / 9781108921183 (cited on page 7). [GW91] Peter W. Glynn and Ward Whitt. “Departures from Many Queues in Series”. The Annals of Applied Probability 1.4 (1991), 546–572. doi: 10.1214/aoap/1177005838 (cited on page 9). [Gwy20] Ewain Gwynne...

    1991

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