Brownian Last Passage Percolation

• BLPP is a paradigmatic, exactly solvable model of random growth in the KPZ universality class, characterized by energy-maximizing paths in a Brownian environment.
• It provides a continuum limit for discrete last passage percolation models, elucidating scaling exponents (1/3 and 2/3), universal fluctuations, and the emergence of the directed landscape.
• Its integrable structure and duality relations reveal deep connections with random matrix theory, stochastic calculus, and probabilistic line ensemble theory.

Brownian Last Passage Percolation (BLPP) is a paradigmatic, exactly solvable model of random growth in the Kardar-Parisi-Zhang (KPZ) universality class. It describes the statistics and geometry of energy-maximizing (directed) paths, or polymers, in a space-time random environment consisting of independent Brownian motions indexed by discrete line or lattice coordinates. BLPP offers a continuum limit for integrable discrete and semi-discrete last passage percolation models, illuminating core aspects of scaling exponents (1/3 and 2/3), universal fluctuations, Airy processes and the directed landscape, rare event geometry, duality structures, competition interfaces, and the remarkable interplay of random matrix theory, stochastic calculus, and probabilistic line ensemble theory.

1. Model Definition, Constructions, and Integrable Structure

BLPP considers independent two-sided Brownian motions $\{B_r\}_{r\in\mathbb{Z}}$. A (directed, up-right) staircase path $\pi$ from $(m, s)$ to $(n, t)$ consists of a non-decreasing sequence of jump times $(\tau_{m-1}, ..., \tau_n)$, with $\tau_{m-1}=s$, $\tau_n=t$. The last passage time is

$$L_{\mathbf{x}, \mathbf{y}}(\mathbf{B}) = \sup_{\pi:\mathbf{x}\to\mathbf{y}} \sum_{r=m}^n B_r(\tau_{r-1}, \tau_r)$$

and the geodesic is the almost surely unique path realizing this maximum.

The model's integrable structure allows exact formulas for multi-point distributions of passage times via determinantal and Fredholm determinant representations, especially for points-to-line (flat) and narrow wedge geometry with functional boundary data. For a deterministic boundary $X:[0,1]\to\mathbb{R}$, the last passage value is

$$BLPP(X;(t,m)) = \max_{0\le t_0\le \cdots \leq t_m=t} X(t_0) + \sum_{k=1}^m [B_k(t_k) - B_k(t_{k-1})].$$

The joint distribution across $k$ space-time points $\{(t_i, m)\}$ is given by a Fredholm determinant of an explicit kernel involving generalized heat kernels, parabolic hitting operators, and boundary data (Rahman, 28 Sep 2024).

BLPP also arises as the scaling limit of geometric/exponential last passage percolation and as an extreme-zero "temperature" $\beta \to \infty$ limit of the O'Connell–Yor/Laplace–log-gamma polymer models (FitzGerald et al., 2019).

2. Scaling Theory, Regularity, and the Directed Landscape

BLPP is a canonical object in the 1+1d KPZ class, exhibiting:

• Longitudinal scaling: For passage times from $(0,0)$ to $(n,n)$, the mean grows as $O(n)$, with fluctuations of order $n^{1/3}$ (the ``1/3 exponent'').
• Transversal fluctuations: Geodesics typically deviate from the straight line by $n^{2/3}$ ("2/3 exponent").

The top line of the canonical BLPP line ensemble converges under parabolic rescaling to the Airy process; more generally, the Airy line ensemble arises as the scaling limit of the multi-line BLPP process (Dauvergne et al., 2018).

The continuum scaling limit, the directed landscape, $L(x,s;y,t)$, is a random function on quadruples $(x,s;y,t)\in\mathbb{R}^2\times\mathbb{R}^2$ with $s<t$, characterized by independent Airy sheet marginals, the metric composition (semigroup) law,

$$L(x,r;y,t) = \max_{z\in\mathbb{R}} \{ L(x,r;z,s) + L(z,s;y,t) \}$$

and stationarity/scale-invariance. The convergence of BLPP to the directed landscape is established in (Dauvergne et al., 2018), and the convergence of (scaled) geodesics to random Hölder $2/3^-$ continuous paths is proved, completing the construction of the universal KPZ-fixed-point object.

Local Regularity and Absolute Continuity

The BLPP polymer weight profile in scaled coordinates enjoys local $1/2$-Hölder continuity with sharp poly-logarithmic corrections,

$$|W(y) - W(x)| \leq C |y-x|^{1/2} (\log |y-x|^{-1})^{2/3}$$

with high probability, uniformly for a broad class of initial data (Hammond, 2017). After affine adjustment, each interval-restricted segment is absolutely continuous with respect to the Brownian bridge law, with the Radon-Nikodym derivative uniformly in $L^p$ for all $p<3$; for the narrow wedge case, the power can be taken arbitrarily large (Hammond, 2017, Tassopoulos et al., 23 Sep 2025).

3. Geodesic Geometry: Coalescence, Disjointness, and Exceptional Structures

Coalescence and Disjoint Polymer Exponents

The probability that $k$ disjoint polymers of unit length start and end close (within $\epsilon$) is

$$P(\text{exist } k \text{ disjoint polymers}) \leq \epsilon^{(k^2-1)/2 + o(1)}.$$

The exponent $(k^2-1)/2$ quantifies the rarity of multi-polymer non-coalescence and encodes the rapidly branching, tree-like structure of geodesics (polymer web) in the scaling limit (Hammond, 2017). For $k=2$, the sharpness of this exponent controls the Hausdorff $1/2$-dimensional set of exceptional points (where non-coalescence occurs) in the Airy sheet [BGH].

The two-thirds power law is also rigorously established for transversal polymer fluctuations: a polymer of short duration $\epsilon$ typically fluctuates by $\epsilon^{2/3}$ from its interpolating line.

Duality of Coalescence and Exit Points

Semi-infinite geodesics in BLPP with fixed direction $\theta$ are constructed using Busemann functions, which are well-defined for all points and all directions, with explicit Burke--queueing structure and monotonicity (Seppäläinen et al., 2021).

A key duality, first rigorously developed for exponential LPP, relates the tail of coalescence times for semi-infinite geodesics to probabilities for exit point measures in stationary models (Pimentel, 2013). In the scaling limit,

$P(T_m < n) = P(Z_n([ -m, m]) = 0)$

where $T_m$ is the coalescence time and $Z_n$ the measure of exit points. The tail of rescaled coalescence times is controlled by the extremal location law of $\sqrt{2}B(u)+\mathcal{A}(u)-u^2$, uniting Brownian and Airy process variational structure.

Coalescence time distributions exhibit heavy tails with non-integrable power law decay, indicating fractal, non-trivial geometry of the limiting polymer web.

4. Temporal Correlations, Line Ensemble Structure, and Phase Transitions

Temporal Correlation Scaling

The decay rate of correlations between passage times at different endpoints encodes the spatial–temporal geometry of the BLPP environment. For line-to-point LPP (flat initial data), the covariance between $X_r$ and $X_n$ behaves as

$$\mathrm{Cov}(X_r, X_n) = \Theta\big( (r/n)^{4/3+o(1)} n^{2/3} \big).$$

Notably, the temporal exponent $4/3$ exceeds the droplet (narrow wedge) case ($2/3$), reflecting the geometry imposed by initial data and the prevalence (or lack) of path overlap (coupling) (Basu et al., 2019). The Brownian Gibbs property and a detailed comparison to Brownian motion are crucial technical tools in establishing these exponents.

Phase Separation in Boundary-Induced Regimes

BLPP-adjacent models with "supercritical" boundary conditions (e.g., half-space geometric LPP with diagonal drift) display a curve separation phenomenon: the top curve decorrelates from the lower ensemble, transitioning from Tracy-Widom to Brownian ($N^{1/2}$) fluctuations (Dimitrov et al., 8 Oct 2025). Remaining curves converge (after $N^{2/3}$/$N^{1/3}$ scaling) to the Airy line ensemble, confirming universality and providing a rigorous prototype for phase transitions in KPZ systems with strong boundary fields.

5. Dynamical BLPP, Stability, Chaos, and Shock/Instability Structure

Dynamical BLPP and Noise Sensitivity

When the BLPP environment is dynamically perturbed (via, e.g., Ornstein-Uhlenbeck flow or discrete resampling), geodesics display a sharp phase transition in overlap: for resampling time $t\ll n^{-1/3}$, the overlap is order $n$ (stable), while for $t\gg n^{-1/3}$, the overlap collapses (chaotic regime) (Ganguly et al., 2020, Bhatia, 31 Oct 2025). The exponent $1/3$ is universal for this "chaos onset". The expected number of macroscopic ("coarse-grained") switches of the geodesic in $[s,t]$ is $n^{5/3+o(1)}(t-s)$, matching the KPZ scaling for potential switch regions. Times at which a non-trivial bigeodesic exists have Hausdorff dimension at most $1/2$ (zero for any fixed direction), confirming the scarcity of exceptional rigid structures.

Shocks, Instability Points, and Competition Interfaces

BLPP naturally encodes notions from stochastic Hamilton-Jacobi (SHJ) equations. For almost all directions, there is a unique eternal solution; for a dense, random, countable set of exceptional directions, multiple solutions exist, differing at instability points (Rassoul-Agha et al., 10 Jul 2024).

• Shock points: Points where two semi-infinite geodesics ("characteristics") for the same Busemann function immediately split but later coalesce. Shock trees (coalescing trees in the primal lattice) arise for each eternal solution.
• Instability region: The subset of space-time where multiple eternal solutions differ is structurally a "web" in the dual lattice, constructed of maximal intervals and vertical edges. The structure is reconstructible from the union of shock trees. This analysis links discontinuities of velocity fields, shock coalescence, and instability web geometry in a KPZ setting.

6. Absolute Continuity, Regularity, and Connections to Random Matrix Theory

Radon-Nikodym Bounds

For BLPP with inhomogeneous (non-decreasing) initial data, the spatial increments' law is absolutely continuous with respect to the Wiener measure, and the $L^p$ ($p>1$) norms of the Radon-Nikodym derivative are controlled as $O(\exp(d_p m^2 \log m))$ for $m$ lines (or curves), with optimal $\exp(d_p m^2)$ growth in the homogeneous case (the top curve of Dyson Brownian motion) (Tassopoulos et al., 23 Sep 2025).

These bounds confirm the robust regularity of BLPP line ensembles (top lines, interface profiles, etc), provide the foundation for regularity properties anticipated for the KPZ fixed point and directed landscape, and connect polynomial growth of moments to random matrix analogues.

Duality to Interacting Particle Systems

Through rigorous determinantal identities and scaling limits, BLPP with arbitrary functional boundaries is connected—via duality transformations—to TASEP-like interacting particle systems. The law of multi-point BLPP observables is described by Fredholm determinants of explicit integral kernels (incorporating heat, hitting, and reflection), which extend the integrable framework to arbitrary deterministic initial data (Rahman, 28 Sep 2024). These formulas subsume previous special cases and allow effective translation of random growth, polymer, and exclusion process insights.

Point-to-line BLPP distributions are identified as invariant laws of reflected (multi-dimensional) Brownian motion systems and connect to random matrix models, including the supremum of Dyson Brownian motion with drift and the Laguerre Orthogonal Ensemble (FitzGerald et al., 2019).

7. Summary Table: Key Structural and Scaling Properties

Feature / Observable	Formula / Exponent / Limit Law	Source(s)
Passage time $\sim$ from $(0,0)$ to $(n,n)$	Mean: $O(n)$, fluctuation: $O(n^{1/3})$	KPZ scaling
Geodesic transversal fluctuation	$O(n^{2/3})$	(Hammond, 2017)
Multi-polymer disjointness exponent	$\epsilon^{(k^2-1)/2}$, sharp for $k=2$	(Hammond, 2017)
Covariance decay, flat initial data	$(r/n)^{4/3} n^{2/3}$	(Basu et al., 2019)
Radon-Nikodym $L^p$ norm, $m$-line case	$O(\exp(d_p m^2 \log m))$ (inhomogeneous), $O(\exp(d_p m^2))$ (homogeneous)	(Tassopoulos et al., 23 Sep 2025)
Coalescence time exponent	$T_m \sim c m^{3/2}$	(Pimentel, 2013)
BLPP scaling limit	Directed landscape, Airy line ensemble, Airy sheet	(Dauvergne et al., 2018)
Dynamical noise-sensitivity	Crossover at $t_c = n^{-1/3}$	(Ganguly et al., 2020)
Bi-infinite geodesic exceptional times	Hausdorff dim $1/2$ ($0$ if direction fixed)	(Bhatia, 31 Oct 2025)

Conclusion

BLPP encapsulates, in a continuum setting, the canon of the KPZ universality class: exact scaling, intricate geometry of geodesics and interface fluctuations, sharp regularity and rare-event behavior, and a deep correspondence between polymers, random matrices, exclusion processes, and stochastic control. Its proof-scale description of coalescence, shocks, instabilities, and dynamical sensitivity, realized through precise probabilistic and integrable tools, provides a model framework for stochastic growth, random environments, and the mathematical structure underlying universality in out-of-equilibrium statistical physics.