Negative association of Busemann functions in exponential last-passage percolation

Technical Summary: Negative Association of Busemann Functions in Exponential Last-Passage Percolation

1. Problem Statement and Motivation

In the study of random growth models within the Kardar–Parisi–Zhang (KPZ) universality class, a central challenge is establishing the universality of fluctuation statistics. For exactly solvable models, such as exponential last-passage percolation (LPP), the existence of product-form invariant measures often allows for the confirmation of KPZ fluctuation exponents ($\chi = 1/3, \xi = 2/3$) via probabilistic arguments relying on the independence of increments.

However, in general settings, Busemann functions (defined as limits of passage time differences) serve as proxies for invariant measures. While Busemann increments are independent when considered along a single direction or under specific monotonic ordering of direction parameters, this independence fails when multiple asymptotic directions are considered simultaneously. The joint invariant measures are not jointly product-form.

The paper addresses the nature of this dependence. While it is conjectured that Busemann increments exhibit negative correlations (which would imply diffusive scaling for sums of increments, a key step in proving $\xi \ge 2\chi$), rigorous evidence has been scarce. Previous work by Alevy and Krishnan established negative correlation for adjacent increments under general weights but excluded the exponential case due to independence in that specific setting. This paper aims to characterize the dependence structure of Busemann increments across arbitrary directions in the exponential LPP setting, specifically investigating whether they exhibit negative association.

2. Methodology

The authors employ a combination of queueing-theoretic representations of LPP and combinatorial properties of queueing maps. The core methodology involves:

• Queueing Map Formalism: The paper utilizes upward ($\sigma_m$) and downward ($\sigma_m$) queueing maps acting on environments of weights. These maps are deterministic transformations that relate LPP with boundary conditions to the bulk LPP.
• Burke Property: A crucial ingredient specific to exponential weights is the Burke property (Proposition 2.15), which states that for $M/M/1$ queues, the output process is independent of the queue length and preserves the exponential distribution, merely swapping rates. This allows the authors to treat the transformation of weight rates as a permutation.
• Braid Relations and Monotonicity: The proof relies heavily on the braid relations of the queueing maps (Proposition 2.13) and their monotonicity properties (Proposition 2.14). These allow the authors to manipulate the order of operations and compare environments.
• Distributional Equivalence: The authors leverage a known result (Proposition 4.1, citing [29]) that establishes a distributional equality between the Busemann process in the bulk and the increments of an LPP model with specific boundary conditions (increment-stationary LPP).
• Reduction to Boundary LPP: By rotating the model and utilizing the distributional equivalence, the problem of analyzing Busemann functions is reduced to analyzing increments in a boundary LPP model with a south boundary.
• Harris-FKG Inequality: The core of the negative association proof involves showing that a specific increment (associated with a "boundary" edge) is negatively correlated with other increments. This is achieved by demonstrating that decreasing the weight of the boundary edge increases (or leaves unchanged) the other increments, while the boundary increment itself decreases. This monotonicity allows the application of the Harris-FKG inequality to establish negative covariance.

3. Key Contributions and Results

Main Theorem: Negative Association (Theorem 1.1)
The primary result establishes that Busemann increments across arbitrary directions are negatively associated. Specifically, for a set of increments $\{B_i\}$ along a down-right path and a set of Busemann functions $\{R_j\}$ defined on a down-right path within the rectangle spanned by the $B_i$'s, the joint distribution satisfies:
$$\mathbb{E}\left[ g(R_1, \dots, R_k) \prod_{i=1}^n f_i(B_i) \right] \le \mathbb{E}\left[ g(R_1, \dots, R_k) \right] \prod_{i=1}^n \mathbb{E}\left[ f_i(B_i) \right]$$
for any non-negative, coordinate-wise monotone functions $f_i$ and $g$ in the same direction. This implies that increasing the values of the $B_i$'s tends to probabilistically decrease the values of the $R_j$'s.

Corollaries:

• Domination of Moment Generating Functions (Corollary 1.3): The joint moment generating function of Busemann increments along a down-right path is dominated by the product of their marginal moment generating functions.
• Exponential Concentration (Corollary 1.4): Even without independence, sums of Busemann increments on the diffusive scale ($\sqrt{n}$) satisfy exponential concentration inequalities. This confirms that fluctuations scale diffusively, supporting the lower bound $\xi \ge 2\chi$.
• Negative Dependence (Corollary 1.5): The increments satisfy negative orthant dependence, meaning the probability of all increments being simultaneously large (or small) is bounded by the product of their marginal probabilities.
• Improved Independence (Theorem 1.8): The paper refines previous independence results. It shows that a Busemann increment $B^\lambda_{a,b}$ is independent of a $\sigma$-algebra generated by increments with direction parameters $\rho \ge \lambda$ (lying "above" the edge) or $\rho \le \lambda$ (lying "below" the edge), provided the spatial ordering is respected. This generalizes previous results that required a totally ordered sequence of parameters along a single path.

Proposition 1.10:
The paper provides an explicit example showing that the correlation is strictly negative (not zero) for non-monotone direction parameters, confirming that the negative association is a non-trivial structural feature.

4. Significance and Claims

The paper claims that the failure of independence in the joint Busemann process is "one-sided," manifesting as negative association rather than arbitrary dependence.

• Implications for KPZ Exponents: The results provide rigorous footing for the heuristic that negative correlations imply diffusive scaling of Busemann fluctuations. This is a necessary component for proving the KPZ relation $\xi = 2\chi$ (specifically the $\xi \ge 2\chi$ direction) in general models, even though the paper focuses on the exactly solvable exponential case.
• Generality of Proof Techniques: While the main result relies on the exponential distribution via the Burke property, the authors emphasize that the "bulk" of the proof (involving hidden LPP monotonicities and braid relations for queueing maps) holds for arbitrary weight distributions. This suggests that a version of the negative association result may extend beyond exactly solvable settings.
• Structural Insight: The work clarifies the complex joint law of Busemann functions, showing that while they are not independent, they possess a strong form of negative dependence that is sufficient for deriving concentration inequalities.

The paper does not claim to solve the full KPZ universality problem for non-solvable models but positions its findings as a significant step toward understanding the correlation structure required to establish fluctuation exponents in those broader contexts.