van den Berg-Kesten--type correlation inequalities for disjoint polymers in the KPZ universality class

TL;DR

This work establishes a BK-type inequality in the positive-temperature KPZ framework, covering the KPZ line ensemble and the continuum directed random polymer by first proving a discrete BK bound for the integrable log-gamma polymer and then passing to the continuum via careful scaling. The key ideas revolve around an extended invariance identity that reexpresses disjoint polymer partition functions as line-ensemble partition functions, together with negative association arguments, to obtain entropy-corrected bounds. The results yield sharp upper-tail control for the KPZ equation and justify convergence phenomena like Brownian-bridge limits under upper-tail conditioning, while highlighting integrability as a crucial ingredient via explicit counterexamples in non-integrable models. The paper also sketches generalizations to more endpoints and higher-indexed curves, outlining the challenges and potential extensions to the CDPR and KPZ_t line ensembles.

Abstract

In classical percolation theory, the van den Berg-Kesten (BK) inequality is a fundamental tool that shows that disjoint events induce negative conditionings on each other. The inequality also holds in the context of last passage percolation (LPP), which is the zero temperature limit of polymer models and an important subclass in the Kardar-Parisi-Zhang (KPZ) universality class. Recently, an analog of the BK inequality was discovered in the context of zero temperature line ensembles and the scaling limit of LPP, where it was used to study upper tail probabilities of the weight and the scaling limit of geodesics under such upper tail conditionings. However, while it has become apparent that such an inequality in the positive temperature setting would have a number of applications, it seems likely that a direct generalization of the zero temperature inequality would not hold. In this work we prove a version of the BK inequality for the KPZ line ensemble and the continuum directed random polymer. We do so by working with the log gamma polymer, making use of its integrability and the geometric RSK correspondence. Our inequality serves as a key input in analyzing the KPZ line ensemble and proving sharp upper tail estimates of the KPZ equation in arXiv:2208.08922, and proving convergence of the continuum directed random polymer to Brownian bridge under the upper tail event in arXiv:2311.12009. The crucial role of integrability in the validity of such an inequality is highlighted via a counter-example for a non-integrable model.

Figures (5)

• Figure 1: The left and middle panels depict the endpoints whose partition functions define the left and right line ensembles (black vertices), respectively, in the right panel. Proposition \ref{['p.extended invariance']} says that the partition function in the original environment when the starting point is on the bottom line and the ending point is on the top or right boundaries equals a certain partition function in the combined line ensemble depicted on the right (some paths which contribute to the latter are shown). Note that the two line ensembles are connected by an auxiliary column of vertices shown in white; these vertices have associated weights which prevent a certain double counting from appearing in the weights of certain paths in the joint line ensemble, which is needed for Proposition \ref{['p.extended invariance']}.
• Figure 2: Left: the orange lines connect the common starting point and varying ending points of the paths whose partition functions determine the values of $Z^{m,n}_j(i)$ for $(i,j)\in J_1[m,n]$, and therefore also of $\{Y_{\mathbf v}\}_{\mathbf v\in V_1[m,n]}$. Right: the blue lines connect the varying starting points and common ending point of the paths whose partition functions determine the values of $Z^{m,n}_j(i)$ for $(i,j)\in J_2[m,n]$, and therefore also of $\{Y_{\mathbf v}\}_{\mathbf v\in V_2[m,n]}$. Note that in both cases the partition functions of the paths joining $(1,1)$ and $(m,n)$ are included, which implies that $Z^{m,n}_j(m) = Z^{m,n}_j(m+1)$ for all $j\in\llbracket1,n\rrbracket$.
• Figure 3: A depiction of the coordinates in Proposition \ref{['thm:invnyc']}.
• Figure 4: A depiction of the graph underlying the line ensembles. Up to swapping the $y$-coordinate $j$ by $n+1-j$, these also depict the coordinates associated to the index sets $J_1[m,n]$ and $J_2[m,n]$, respectively. The paths in red, blue, and orange are examples of the paths contributing to the partition function $S([(a,1)^{\shortuparrow}], [(m+b,b)])$; they go up-right in the left line ensemble, and down-right in the right line ensemble.
• Figure 5: The purple paths are the only ones which contribute to $S([(1,1)^{k-1,\shortuparrow}], [(m,n)]^{k-1})$ (here, $k=3$). In $S([(1,1)^{k-1,\shortuparrow}, (a,1)^{\shortuparrow}], (m,n)^{k})$, the same purple paths are frozen, and there is an additional factor of the partition function associated to the starting and ending points $(a,1)^{\shortuparrow}$ and $(m,n-k+1)$ (the weight of the depicted orange path is one contribution to it). Thus in the ratio of the two terms the contribution of the top $k-1$ frozen paths cancels.

Theorems & Definitions (67)

• Remark 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.3
• Theorem 1.4
• Remark 1.5: Terminology
• Theorem 2.1
• Remark 2.2
• Corollary 2.3
• proof