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Duality between coalescence times and exit points in last-passage percolation models. (English) Zbl 1361.60095
The paper under review proves a duality relation between coalescence times and exit points in last-passage percolation models with exponential weights. The exact scaling behavior of the coalescence times is fundamental, and is conjectured to have scaling exponent 3/2. It is expected to converge to an universal limiting distribution under the scaling exponent 3/2.
Let \(\omega = \{W_x: x\in \mathbb Z^2\}\) be a set of i.i.d random variables with exponential distribution function of parameter one, where \(W_x\) is the passage (or percolation) time through the vertex \(x=(x(1), x(2))\) in the lattice \(\mathbb Z^2\). Let \(\Gamma (x, y)\) be the set of all up-right oriented paths \(\gamma = (x_0, x_1, \cdots, x_k)\) from \(x=x_0\) to \(y=x_k\) with \(x_{j+1}-x_j\in \{e_1, e_2\}\) for \(j=0, 1, \cdots, k-1\). The passage time of \(\gamma\) is defined to be \(W(\gamma) = \sum_{j=1}^kW_{x_j}\), and the last-passage time between \(x\) and \(y\) is defined to be \(L(x, y) = \max_{\gamma \in \Gamma (x, y)}W(\gamma)\). The geodesic path from \(x\) to \(y\) is the a.s. unique maximizing path \(\gamma (x, y)\) with \(L(x, y) = W(\gamma (x, y))\). The first coalescence point in the up-right orientation \(c(x, y)\) is to be understood in the sense that \(c(x, y) \leq c^{\prime}\) for every other geodesic point \(c^{\prime}\in Z^2\) with \(\gamma (z) = \gamma (z, c^{\prime}) + \gamma (c^{\prime})\) for \(z=x, y\). The coalescence time \(T_m\) is the second coordinate of \(c(me_1, me_2)\) for \(m\geq 1\). The exit point counting measure process is \(Z_n = (\zeta_n (z))_{z\in \mathbb Z} \in \{0, 1\}^{\mathbb Z}\), where for fixed \(n\geq 1\) the term \(\zeta_n (z) =1\) if \(z= Z(x, n)\) for some \(x\in [1, \infty)\) and zero otherwise.
The main result of the paper is Theorem 1 (duality formula), which shows that \[ P(T_m< n) = P(Z_n ([-m, m])=0), \] for \(n> m>0\). Section 2 begins with the last-passage percolation model and the exclusion process, and shows the Burke property for the Busemann function (Lemma 1), where the Busemann function \(B(x, y) = L(c, y) - L(x, c)\) has a certain distribution property on the integer lattice. Then the author proves the self-duality property for the geodesic tree (Lemma 2). The proof of Theorem 1 follows from the equivalence between the stationary LPP model with boundary, and the Busemann functions allow to interpret exit points as crossing points of semi-infinite geodesics. The coalescence time \(T_m^*\) of the dual trees and the crossing points process are the same by a topological consequence and Lemma 2.
The tail distribution of the coalescence time is defined to be \[ G(r) = \lim \inf_{m\to \infty} P(\frac{T_m}{2^{-5/2}m^{3/2}}>; r). \] Theorem 2 shows that the tail distribution satisfies \(G(r) \geq 1 - c_0r^2\) and \(\lim_{r\to 0^+} G(r) =1\) for some constants \(c_0\) and \(r_0 >0\) and \(r\in [0, r_0]\). The proof of Theorem 2 is given in subsection 2.6 by using the previous duality formula from Theorem 1. The last-passage time has a variational representation, and the variational representation for exit points, together with the scaling limit of the last-passage times, implies a limit theorem for \(Z_n\) (Lemma 4). Theorem 3, stating that for \(r>0\), \(G(r) \geq F(r^{-2/3}) - F(- r^{-2/3})\), is proved by Lemma 4.
Section 3 states a conjecture for the scaling limit of the coalescence times. The author conjectures that the tail distribution is given by \(G(r) = P\left(U(-r^{-2/3}, r^{-2/3}]\geq 1\right)\) for a counting process \(U\) composed by a Dirac measure located at \(U(v)\), given by the Busemann function and the \(\text{Airy}_2\) sheet.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60C05 Combinatorial probability
60F05 Central limit and other weak theorems
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