Geometric Last Passage Percolation

Updated 21 October 2025

Geometric LPP is an integrable random growth model on a planar lattice that assigns independent geometric weights to vertices and computes maximal up/right path sums.
The model exhibits KPZ scaling with fluctuations of order N^(1/3) and transversal deviations of order N^(2/3), leading to Tracy–Widom limit laws in typical settings.
Its determinantal structure links the model to random matrix theory, enabling precise analysis of multi-point correlations, rare events, and geodesic coalescence.

Geometric Last Passage Percolation (LPP) is a paradigmatic integrable random growth model defined on the planar lattice, where independent geometric random variables are assigned to the vertices (or edges), and the central object is the maximal weight collected by up/right (directed) lattice paths between given points. This model lies in the Kardar–Parisi–Zhang (KPZ) universality class, and, due to its exact solvability, it provides a rich testing ground for investigating scaling exponents, limit distributions, multi-point and multi-time correlation functions, geometry of geodesics, large deviations, rare event tails, and related topics. Geometric LPP admits a determinantal structure and connections to random matrix theory, combinatorial representation theory, and integrable probability.

1. Model Definition and Integrable Structure

The geometric LPP model is specified by assigning independent geometric random variables $w(i,j)$ (typically $\mathrm{Geom}(q^2)$ , $q \in (0,1)$ ) to each vertex $(i,j)$ in $\mathbb{Z}_{>0}^2$ . The last-passage time to a point $(m, n)$ is

$G(m,n) = \max_{p\in \mathcal{P}} \sum_{(i,j)\in p} w(i,j)$

where the maximum is over all up/right lattice paths $p$ from $(1,1)$ to $(m,n)$ . The distribution of $G(m,n)$ admits exact contour integral representations, Fredholm determinant (and, for symmetrized/half-space variants, Pfaffian) formulas, and strong duality with random matrix ensembles (notably the Jacobi and truncated unitary ensembles) (Byun et al., 20 Oct 2025).

Integrability enables computation of multi-point, multi-time, and stationary distributions, as well as asymptotic analysis via steepest descent, analytic continuation, and operator theory. These features fundamentally distinguish geometric LPP from non-integrable directed percolation models.

2. KPZ Scaling, Fluctuations, and Limit Laws

Geometric LPP exhibits scaling exponents and fluctuation properties characteristic of the KPZ universality class. The typical last-passage time for $(m,n)=([v_1 N],[v_2 N])$ grows linearly with $N$ , with fluctuation scale of order $N^{1/3}$ , and geodesic (optimal path) transversal fluctuations of order $N^{2/3}$ . After centering and scaling, $G(m,n)$ converges in distribution to the Tracy–Widom GUE law in typical settings. For half-space or certain boundary conditions, limiting laws are Tracy–Widom GOE, GSE, or Baik–Rains distributions, depending on the regime and geometry (Betea et al., 2019, Zeng, 16 Jul 2025, Betea et al., 22 Jul 2025).

Large deviation analysis shows a sharp asymmetry between upper and lower tails: probabilities of atypically small values of $G_{n,m}$ decay as $\exp(-\Theta(N^2))$ , while upper tail deviations decay as $\exp(-\Theta(N))$ , with precise formulas (including subleading terms) available via duality with random matrices (Byun et al., 20 Oct 2025). The shape and fluctuation exponents are robust under mild perturbations of the weight distribution, but not in certain “hierarchical” or heavy-tailed regimes, where logarithmic corrections and new critical exponents appear (Ganguly et al., 12 Nov 2024).

3. Multi-Point and Multi-Time Correlations

One of the most significant features of the geometric LPP is the exact (determinantal) structure enabling analysis of joint fluctuations at multiple points or times. For the two-time distribution, consider $G(m_1,n_1)$ and $G(m_2,n_2)$ with $m_1<n_1<m_2<n_2$ ; after centering and scaling in the KPZ $1:2:3$ regime, the joint law converges to a universal limit expressible as a contour integral of a Fredholm determinant (Johansson, 2018). The matrix kernel is built from explicitly constructed block kernels (e.g., Airy functions), and the combination of complex analysis and operator theory allows for evaluation of universal correlation functions capturing time-like dependences (Basu et al., 2018).

In half-space and stationary variants, explicit Fredholm Pfaffian formulas are derived, with critical scaling limits interpolating between Baik–Rains and other KPZ-class distributions depending on the boundary parameters (Betea et al., 2019, Zeng, 16 Jul 2025). The analysis crucially involves analytic continuation and steepest descent asymptotics of contour integrals.

4. Geometry of Geodesics and Coalescence Phenomena

The geometry and statistics of geodesics (maximizing paths) in geometric LPP are a central topic of paper. In the homogeneous model, almost surely, for each non-axial direction, all semi-infinite geodesics coalesce, forming a unique directed geodesic tree (with a single “end” at infinity) (Balázs et al., 2023, Janjigian et al., 2019). The backward sub-tree of a fixed vertex exhibits power-law tail behavior in depth ( $n^{-2/3}$ ) and volume ( $n^{-2/5}$ ), with matching upper and lower bounds up to constants (Balázs et al., 2023). These results are consequences of sharp control on transversal fluctuation scales and coalescence theorems.

A central object in the analysis of geodesics is the Busemann function, defined as the almost sure limit of passage time differences along rays in given directions; these satisfy cocycle and ergodic properties and are deeply linked to the stationary versions of LPP. In the exponential model, the Busemann process provides a complete description of the uniqueness, coalescence, and multiplicity of infinite geodesics, with the geometry of “instability points”—locations where geodesic direction jumps—revealing connections to renewal processes and symmetric random walks (Janjigian et al., 2019).

The non-existence of non-trivial bi-infinite geodesics (i.e., infinite paths optimal between any two of their segments) is rigorously proved in geometric LPP based on increment-stationary arguments and explicit tail bounds on exit points from the boundary (Groathouse et al., 2021). These arguments rely on the structure of the increment-stationary model, additivity of passage times given suitable boundary conditions, and sharp large deviation/excursion estimates; the approach extends to non-integrable models provided two key assumptions (exit point decay, random walk bounds) are verified.

In dynamical LPP, where the underlying environment is perturbed (e.g., via Ornstein–Uhlenbeck dynamics), “exceptional times” when bi-infinite geodesics appear have very thin (at most $1/2$) Hausdorff dimension (Bhatia, 16 Apr 2025). Semi-infinite geodesics are proven to be well-directed, with no non-trivial axially-directed paths possible, for all times.

5. Phase Transitions, Stationary States, and Boundary Effects

Boundary conditions and the geometry of the domain play a pivotal role in geometric LPP. On strips and half-spaces, stationary solutions and their limiting distributions are classified via explicit formulas for stationary measures as functions of boundary parameters (e.g., “density,” reservoir strengths) (Bryc et al., 2 Jun 2025, Zeng, 16 Jul 2025).

On a finite-width strip, the stationary measure can be described via free Askey–Wilson functionals, yielding explicit integral formulas for the multipoint generating function for the stationary distribution, and producing a phase diagram analogous to that of the open ASEP (Bryc et al., 2 Jun 2025). Different domains (lower triangular, truncated shapes) yield distributions governed by distinct Tracy–Widom laws (GOE, GSE) or their generalizations (Betea et al., 22 Jul 2025). In the half-space, the last-passage time distribution along the diagonal is governed by a Fredholm Pfaffian structure, with a two-parameter phase diagram dictated by boundary data, and in the supercritical regime, a phase separation arises between top curves (Brownian) and lower curves (Airy line ensemble) (Dimitrov et al., 8 Oct 2025).

These results highlight that the large-scale asymptotics and microscopic statistics depend sensitively on both integrable structure and choice of domain/boundary.

6. Large Deviations and Rare Event Probabilities

Geometric LPP admits a detailed and precise large deviation theory. For general weights with exponential moments and strictly convex shape function, the asymptotics of the probability that the geodesic’s midpoint deviates by distance $tn$ from its typical location are given in terms of the right-tail large deviation rate function for the last-passage value and the shape function, leading to explicit exponential rate asymptotics (Alberts et al., 2 Feb 2025). For geometric weights, the lower and upper tail probabilities of the last-passage time are characterized up to, and including, constant terms: $\log P(G_{n,m} \leq \delta N) = L_1 N^2 + L_2 N + L_3 \log N + L_4 + o(1)$ for the lower tail, and

$\log P(G_{n,m} \geq \delta N) = U_1 N + U_2 \log N + U_3 + o(1)$

for the upper tail (Byun et al., 20 Oct 2025). These refined expansions are derived via duality with the Jacobi Unitary Ensemble and the Truncated Unitary Ensemble, and connect LPP rare event statistics directly to random matrix theory (e.g., moments of the modulus of the characteristic polynomial).

In critical hierarchical or heavy-tailed environments, the last-passage time exhibits polylogarithmic corrections to linear growth, with exponents (e.g., $(\log n)^{3/4}$ for i.i.d. tail exponent 2; $(\log n)^{1/2}$ for branching random walk) that fundamentally depart from classical KPZ scaling (Ganguly et al., 12 Nov 2024).

7. Generalizations, Variations, and Future Directions

The geometric LPP framework has been extended in multiple directions: (i) constraints on admissible paths (e.g., Hölder or entropy constrained LPP, which interpolate between classical models and non-directed or polymer settings) (Berger et al., 2018), (ii) growth on cylinders and with feedback-dependent waiting times (generalized LPP), with explicit invariant laws on the front line derived via probabilistic cellular automata (Casse, 2019), and (iii) multi-path (“watermelon”) settings, where the scaling exponents for collections of disjoint geodesics display determined $k$ -dependence (weight fluctuations $k^{5/3} n^{1/3}$ , transversal $k^{1/3} n^{2/3}$ ), governed by an interlacing property (Basu et al., 2020).

Research continues into scaling limits (directed landscape), universality beyond integrable models, large deviation principles at the metric and geodesic level (Agarwal, 24 Apr 2025), and the geometry of geodesic trees, intersection properties, and path probabilities in more general and dynamically evolving settings (Balázs et al., 2023, Bhatia, 16 Apr 2025).

Table: Central Results and Corresponding Objects/Formulas

Phenomenon	Key Formula/Structure	Reference(s)
Two-time joint distribution	$\displaystyle \lim_{T\to\infty} P(a,A) = \frac{1}{2\pi i} \int_{\mathcal{C}} \frac{du}{u-1} \det(I+K(u))$	(Johansson, 2018)
Stationary measure on strip/half-space	Free Askey–Wilson functional, Fredholm Pfaffian, phase diagram	(Bryc et al., 2 Jun 2025, Zeng, 16 Jul 2025)
Fluctuation scale and Tracy–Widom law	$N^{1/3}$ for fluctuations, $N^{2/3}$ for transversal	(Betea et al., 2019, Betea et al., 22 Jul 2025)
Upper/lower large deviations	$\log P(G_{n,m} \leq \delta N) = \Theta(N^2)$ (lower), $\log P(G_{n,m} \geq \delta N) = \Theta(N)$ (upper)	(Byun et al., 20 Oct 2025)
Geodesic coalescence, nonexistence bi-infinite	Increment-stationary model, sharp exit point tail bounds	(Groathouse et al., 2021)
Multi-path (watermelon) exponents	$k^{5/3} n^{1/3}$ (weight), $k^{1/3} n^{2/3}$ (transversal)	(Basu et al., 2020)

Geometric LPP, by virtue of its exact solvability, determinantal structure, and connections to random matrices, is a cornerstone of integrable probability. It provides a rigorous mathematical structure for understanding universal scaling, multi-point spatio-temporal correlations, nontrivial geometry of optimal paths, rare event statistics, and the effects of boundaries and domain geometry. Its relevance extends to random matrix theory, combinatorial representation theory, and statistical mechanical models of growing interfaces.