Last Passage Percolation with Geometric Weights

• Last passage percolation with geometric weights is a stochastic process that assigns independent geometric random variables on lattice vertices to maximize up-right path sums.
• Its integrability links the model to the RSK correspondence, Schur processes, and allows the derivation of exact results through Fredholm determinants and Pfaffians.
• The model underpins KPZ universality by serving as the zero-temperature limit of directed polymers and offering precise insights into scaling exponents, geodesic geometry, and phase transitions.

Last passage percolation (LPP) with geometric weights is a paradigmatic model in integrable probability, stochastic growth, and the Kardar–Parisi–Zhang (KPZ) universality class. In this model, independent geometric random variables are assigned to vertices (or edges) of a planar lattice, and the last passage time between two points is defined as the maximal sum of weights along any up-right path. Geometric LPP admits rich exact structures, including connections to the Robinson–Schensted–Knuth (RSK) correspondence, Schur processes, Fredholm determinants, and Pfaffians. These exact solvability features enable precise probabilistic and combinatorial analysis of passage times, geodesic geometry, limit theorems, fluctuation exponents, and phase transitions in both full-space and restricted geometries. Geometric LPP also serves as the zero-temperature limit of directed polymers with geometric weights and as a foundation for comparison with more general or non-integrable models.

1. Model Definition and Integrable Structure

Let $\{w_{i,j}\}$ be an array of independent geometric random variables with parameter $q$ ($0 < q < 1$), i.e., $\mathbb{P}(w=k) = (1-q)q^k$, $k\geq0$. For fixed indices $(m,n)\in\mathbb{N}^2$, define the up-right (or NE) paths $\pi$ from $(1,1)$ to $(m,n)$, and the last passage time

$G_1(m, n) = \max_{\pi} \sum_{(i,j)\in \pi} w_{i,j}$

where the maximum is over all up-right paths from $(1,1)$ to $(m,n)$. The model generalizes to tuples of disjoint paths via

$$G_k(m, n) = \max_{\{\pi_1, \dots, \pi_k\}} \sum_{l=1}^k \sum_{(i,j)\in\pi_l} w_{i,j}$$

with the $\pi_l$ mutually disjoint. The increments $\lambda_1 = G_1$, $\lambda_k = G_k - G_{k-1}$ for $k\geq 2$ define a partition, forming the "partition coordinates" associated with the point $(m, n)$.

A central feature is the combinatorial equivalence between geometric LPP and the evolution of partitions given by the Schur process, as established via generalizations of the RSK correspondence and Greene's theorem (Dimitrov et al., 6 Oct 2025). The partition sequence along a general down-right path in the LPP is distributed as a Schur process, with weights expressible in terms of skew Schur polynomials and explicit transition measures determined by the geometry and parameters.

2. Exact Distributional Results and Limit Theorems

The integrable nature of geometric LPP allows derivation of explicit distributional formulas in both finite and scaling regimes:

• The distribution of $G_1(n,n)$ and of multi-point passage times can be described via Fredholm determinant formulas and, for certain boundary/half-space problems, via Fredholm Pfaffians (Johansson, 2018, Zeng, 16 Jul 2025). For instance, the two-time scaling limit of LPP passage times at distinct space-time points is given by a contour integral of a Fredholm determinant, with kernels constructed from functions reflecting the model’s integrable (determinantal) structure.
• For higher-rank (multi-path) observables, combinatorial bijections (RSK, Fomin's growth diagrams) and representation-theoretic identities underlie generating functions, yielding exact identities and asymptotic factorizations (Betea, 2018, Dimitrov et al., 6 Oct 2025). Generating series for LPP in point-to-line, point-to-half-line, and half-space-reflected geometries factor as products, and their asymptotics connect to Tracy–Widom GOE and $\mathrm{GOE}^2$ distributions.
• In stationary versions (classical and half-space settings), two-parameter stationary distributions parameterized by $(r, s)$ are characterized, and explicit finite-time formulas for diagonal passage times are given in terms of Fredholm Pfaffians. Under KPZ/critical scaling, the centered and scaled passage times converge to a non-Gaussian, Airy-type limiting law (Zeng, 16 Jul 2025).
• In the one-dimensional analogue and in random graph settings (e.g., Barak–Erdős graph), the LPP constant is characterized through infinite bin models and renewal/skeleton structures, yielding analytic expressions and asymptotic expansions for the limiting longest path per site (Foss et al., 2023, Foss et al., 2020). With geometric weights, strong laws of large numbers and functional central limit theorems are available under appropriate finite moment conditions (Foss et al., 2011).

3. Fluctuation Theory, Scaling Exponents, and Geodesic Geometry

Geometric LPP is a canonical KPZ model with well-understood scaling exponents and fluctuation phenomena:

• The passage time $G_1(n, n)$ grows linearly with $n$ with deterministic "shape function" $W(x, t)$, with fluctuations of order $n^{1/3}$ and transversal (geodesic) fluctuations of order $n^{2/3}$. These exponent values are KPZ-universal and rigorously arise in geometric/exponential LPP and their combinatorial analogues (Johansson, 2018, Basu et al., 2020, Alberts et al., 2 Feb 2025, Cator et al., 2010).
• Large deviation estimates for passage times and geodesic midpoint fluctuations are governed by explicit rate functions, characterized as limits of normalized log-probabilities and directly computed in the exponential case. For geometric weights, analogous results are expected, though discrete effects may require technical adjustments (e.g., unique geodesic selection rules) (Alberts et al., 2 Feb 2025).
• Geodesic structure exhibits strong regularity: in positive-temperature extensions, as well as for zero-temperature geometric weights, there are no non-trivial bi-infinite geodesics (every bi-infinite geodesic is axis-aligned) (Groathouse et al., 2021). Semi-infinite geodesics exist in all directions, are unique, and exhibit coalescence; Busemann functions can be constructed and are linked to equilibrium measures in driven particle systems (Cator et al., 2010, Alevy et al., 2021).
• In multi-path settings, the “geodesic watermelon” object (disjoint path collections maximizing total weight) exhibits: (i) leading order $k\mu n$, (ii) KPZ-exponent refined defect $k^{5/3} n^{1/3}$, (iii) transversal fluctuations $k^{1/3} n^{2/3}$, and (iv) deterministic interlacing and rigidity estimates, extending “1/3–2/3” KPZ theory to multi-path configurations (Basu et al., 2020). Interlacing properties imply strict monotonicity of fluctuations as the number of paths varies.

4. Fredholm (Determinant and Pfaffian) Formulas and Schur Process Equivalence

The probabilistic/combinatorial structure of geometric LPP translates to exact analytic formulae:

• The joint law of LPP values along a general down-right path is distributed as a Schur process:

$$\mathbb{P}(\lambda(v_i) = \lambda_i,\, 0 \leq i \leq M+N) = Z \cdot \mathbf{1}_{\lambda^0 = \lambda^{M+N} = \varnothing} \prod_{i=1}^{M+N} Q(\lambda^{i-1},\lambda^i)$$

where $Q(\lambda,\mu)$ is a (skew) Schur polynomial weight depending on the path’s geometry and parameters (Dimitrov et al., 6 Oct 2025). The proof utilizes Fomin’s growth diagrams and a generalization of Greene’s theorem for Ferrers-shape fillings.

• Finite-time and scaling limit probability distributions (e.g., for $G(n, n)$, joint two-time distributions) take the form of contour integrals over (Fredholm) determinants, with kernels in terms of Airy functions and their discrete analogues (Johansson, 2018). In half-space geometries, Fredholm Pfaffians and corrected kernel formulas arise, and under critical scaling converge to KPZ Airy-type limits (Zeng, 16 Jul 2025).
• These exact formulas are robust to parameter deformations and different boundary/initial conditions. In particular, the transition kernels retain rank-one corrections and shifted discrete derivatives in the limiting picture, with algebraic cancellations ensuring well-defined limiting distributions.

5. Phase Transitions, Stability, and Chaotic Response

Geometric LPP displays robust phase transition behavior and sensitivity under perturbations:

• When random vertex weights are dynamically perturbed (resampled independently with probability $t$), the system undergoes a transition from stability (geodesics and passage times at time 0 and time $t$ highly correlated) to chaos (geodesics decorrelate and become disjoint) as $t$ crosses the threshold $t_*\sim n^{-1/3}$ (planar KPZ case) (Ahlberg et al., 2023). This threshold is set by the scale of the variance of the unperturbed passage time.
• Robust lower bounds on central moments and fluctuations hold more generally, not only in exactly solvable cases. For example, in thin rectangles with aspect ratio $n \times n^\alpha$, $0 < \alpha < 1/3$, the $r$-th central moment is bounded below by $n^{r(1-\alpha)/2}$ (Houdré et al., 2019).

6. Generalizations, Applications, and Connections

• The geometric LPP model serves as a reference point for last passage and greedy path models with more general or heavy-tailed/discrete weight distributions (Chang et al., 2022, Ganguly et al., 12 Nov 2024). The zero-temperature limit of directed polymers with geometric disorder is geometric LPP, and inference techniques developed in the solvable case guide the paper of general environments.
• The time constant or "shape function" is central in both planar and one-dimensional long-range models, with analytic and combinatorial characterizations (including scaling properties, convexity, and lines of non-differentiability) (Foss et al., 2020, Foss et al., 2023).
• Hidden invariance, as revealed via the geometric RSK correspondence, results in rich symmetry properties: shift-invariance, rearrangement invariance, and preservation of multi-point and edge-value joint laws (Dauvergne, 2020). These symmetries extend to universal limits such as the KPZ fixed point and Airy sheet.
• The geometric approach in LPP gives rise to geometric and polyhedral decompositions of the optimization landscape, with domains of optimality for distinct paths forming polyhedral cones, whose volume measures correspond to path probabilities (Alberts et al., 2019).

7. Open Problems and Future Directions

• While the explicit form of rate functions and scaling exponents is established for exponential/geometric weights, precise identification for more general (e.g., heavy-tailed, hierarchical, or non-integrable) environments is open, with recent advances in multi-scale approaches for critical heavy-tail exponents (Ganguly et al., 12 Nov 2024).
• Multi-point and multi-time correlation structures in partially reflected/half-space and multi-path geometries invite further exact analysis, particularly in models with tunable boundary or diagonal weights (Betea, 2018).
• Numerical and simulation methods for the exact computation of LPP constants and distributions (via perfect simulation or finite-state truncations) are refined in geometric settings, where analytic tractability aids in benchmarking and understanding more complex environments (Foss et al., 2023).
• Extending key geometric and combinatorial techniques (e.g., interlacing, rigidity, large deviation functionals) to non-integrable or critical models is an active area, with the geometric LPP framework providing the canonical starting point.

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Key References:

• Distributional equality to Schur processes and RSK/Greene bijections: (Dimitrov et al., 6 Oct 2025)
• Two-time distribution and contour integral formulas: (Johansson, 2018)
• Stationary half-space, Fredholm Pfaffian formulas, and KPZ scaling limits: (Zeng, 16 Jul 2025)
• Fluctuation exponents, rigidity, geodesic watermelons: (Basu et al., 2020)
• Large deviation rate functions for geodesic midpoint: (Alberts et al., 2 Feb 2025)
• Nonexistence of nontrivial bi-infinite geodesics and boundary process approach: (Groathouse et al., 2021)
• Hidden invariance via geometric RSK: (Dauvergne, 2020)
• Geometry of optimization domains: (Alberts et al., 2019)