Intertwining the Busemann process of the directed polymer model

TL;DR

This work advances the study of Busemann processes for planar directed polymers by proving sharp regularity results without assuming differentiability of the shape function, and by giving a complete characterization of the joint Busemann distribution via a Markovian intertwining with a geometric RSK-like dynamics. It develops a two-tier framework that covers general i.i.d. weights and the exactly solvable inverse-gamma case, delivering a tractable description in the solvable regime: Busemann increments on a lattice edge become independent in the inverse-gamma setting and relate to a two-dimensional Poisson process, while the zero-temperature limit recovers the exponential CGM Busemann structure. The paper also connects Busemann discontinuities to competition interfaces, constructs a coupling that yields interface directions from every lattice point, and discusses the breakdown of the one force–one solution principle in the inverse-gamma case through eternal-solution considerations for a discrete stochastic heat equation. Methodologically, it leverages update maps, intertwining identities, and geometric RSK to link polymer dynamics with integrable probabilistic structures, producing invariant measures, ergodicity results, and a rigorous framework for the interface geometry in positive temperature. These results provide a unified, regularity-based understanding of Busemann processes and polymer interfaces across general and solvable settings, with implications for infinite-volume Gibbs measures and disordered KPZ-type equations. The findings illuminate the rich interaction between stochastic growth models, combinatorial correspondences, and ergodic properties of polymer measures, contributing to a deeper grasp of the stochastic heat equation and its discrete analogues.

Abstract

We study the Busemann process and competition interfaces of the planar directed polymer model with i.i.d.\ weights on the vertices of the planar square lattice, in both the general case and the solvable inverse-gamma case. We prove new regularity properties of the Busemann process without reliance on unproved assumptions on the shape function. For example, each nearest-neighbor Busemann function is strictly monotone and has the same random set of discontinuities in the direction variable. When all Busemann functions on a horizontal line are viewed together, the Busemann process intertwines with an evolution that obeys a version of the geometric Robinson-Schensted-Knuth correspondence. When specialized to the inverse-gamma case, this relationship enables an explicit distributional description: the Busemann function on a nearest-neighbor edge has independent increments in the direction variable, and its distribution comes from an inhomogeneous planar Poisson process. The distribution of the asymptotic competition interface direction of the inverse-gamma polymer is discrete and supported on the Busemann discontinuities which -- unlike in zero-temperature last-passage percolation -- are dense. Further implications follow for the eternal solutions and the failure of the one force -- one solution principle of the discrete stochastic heat equation solved by the polymer partition function.

Figures (7)

• Figure 3.1: A sample of all finite polymer paths terminating at $\mathbf{0}$, coupled via \ref{['buqp1']}. The competition interface $\varphi^\mathbf{0}$ is the solid line on the dual lattice $\mathbb{Z}^2+(-\tfrac{1}{2},-\tfrac{1}{2})$. Paths from the west and north of $\varphi^\mathbf{0}$ reach $\mathbf{0}$ through $-\mathbf{e}_1$, while paths from the east and south of $\varphi^\mathbf{0}$ reach $\mathbf{0}$ through $-\mathbf{e}_2$.
• Figure 4.1: A simulated trajectory of the pure jump process $\{B^{\xi(\rho)-}_{x-\mathbf{e}_1,x}\}_{\rho\,\in\, [0,\alpha)}$, with $\alpha=20$. The initial value is $\log W_x \sim \log$ Ga$^{-1}$($\alpha$) and the jumps are determined by an independent Poisson point process on $(0,\alpha)\times\mathbb{R}_{>0}$ with intensity measure $\frac{e^{-y(\alpha-s)}}{1-e^{-y}}\,\mathrm{d} s\, \mathrm{d} y$, according to \ref{['Def:X']}. There are infinitely many jumps on any open interval in $(0,\alpha)$. The process tends to infinity almost surely, as $\rho\nearrow \alpha$.
• Figure 6.1: The form of the $z$ array in the case $m=3$ and $n=5$. The first diagonal is $z_{\bullet 1}=(z_{11}, z_{21}, z_{31}, z_{41}, z_{51})$ and the second one $z_{\bullet 2}=(z_{22}, z_{32}, z_{42}, z_{52})$.
• Figure 6.2: The form of a fully triangular array $z$ in the case $m=n=N=5$. From right to left there are five diagonals $z_{\bullet\ell}=(z_{\ell\ell},\dotsc,z_{5\ell})$ for $\ell=1,2,\dotsc,5$.
• Figure 6.3: Evolution of a triangular array $z(m)$ with $N$ rows and diagonals over time $m=0,1,2,\dotsc$. The initial state $z(0)$ is on the left edge and time progresses from left to right. At time $m$, the driving weights come from row $m$ of the $d$-matrix: $a_1(m)=d_{m\bullet}=(d_{m,1},\dotsc, d_{m,N})$. The update of $z(m-1)$ to $z(m)$ diagonal by diagonal is represented by the downward vertical progression of row insertions. Each cross reduces the length of $a_\ell(m)$ by one and after $N$ steps the last output $a_{N+1}(m)$ is empty.

Theorems & Definitions (59)

• Remark 2.5: Busemann process and a regularity assumption
• Remark 2.6: Discontinuities and null events
• Remark 2.7: Monotonicity
• Remark 3.5: Vertical increments
• Remark 3.6: Polymer-measure interpretation of results on discontinuity set
• Remark 3.8: Relation to competition interface
• Remark 3.9: Comparison with zero temperature, part 1
• Remark 3.11: Comparison with zero temperature, part 2
• Remark 3.12: Open questions
• proof