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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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##### References:
 [1] Aldous, D. and Diaconis, P. (1995). Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theory Related Fields 103 199-213. · Zbl 0836.60107 · doi:10.1007/BF01204214 [2] Baik, J., Ferrari, P. L. and Péché, S. (2010). Limit process of stationary TASEP near the characteristic line. Comm. Pure Appl. Math. 63 1017-1070. · Zbl 1194.82067 · doi:10.1002/cpa.20316 · arxiv:0907.0226 [3] Balázs, M., Cator, E. and Seppäläinen, T. (2006). Cube root fluctuations for the corner growth model associated to the exclusion process. Electron. J. Probab. 11 1094-1132 (electronic). · Zbl 1139.60046 · eudml:129040 · arxiv:math/0603306 [4] Cator, E. and Groeneboom, P. (2006). Second class particles and cube root asymptotics for Hammersley’s process. Ann. Probab. 34 1273-1295. · Zbl 1101.60076 · doi:10.1214/009117906000000089 · arxiv:math/0603345 [5] Cator, E. and Pimentel, L. P. R. (2012). Busemann functions and equilibrium measures in last passage percolation models. Probab. Theory Related Fields 154 89-125. · Zbl 1262.60094 · doi:10.1007/s00440-011-0363-6 · arxiv:0901.2450 [6] Cator, E. and Pimentel, L. P. R. (2015). On the local fluctuations of last-passage percolation models. Stochastic Process. Appl. 125 538-551. · Zbl 1326.60134 · doi:10.1016/j.spa.2014.08.009 · arxiv:1311.1349 [7] Corwin, I., Ferrari, P. L. and Péché, S. (2010). Limit processes for TASEP with shocks and rarefaction fans. J. Stat. Phys. 140 232-267. · Zbl 1197.82078 · doi:10.1007/s10955-010-9995-7 · arxiv:1002.3476 [8] Corwin, I., Quatel, J. and Remenik, D. (2011). Renormalization fixed point of the KPZ universality class. Available at . arXiv:1103.3422 · arxiv.org [9] Coupier, D. (2011). Multiple geodesics with the same direction. Electron. Commun. Probab. 16 517-527. · Zbl 1244.60093 · doi:10.1214/ECP.v16-1656 · arxiv:1104.1321 [10] Ferrari, P. A., Martin, J. B. and Pimentel, L. P. R. (2009). A phase transition for competition interfaces. Ann. Appl. Probab. 19 281-317. · Zbl 1185.60109 · doi:10.1214/08-AAP542 · arxiv:math/0701418 [11] Ferrari, P. A. and Pimentel, L. P. R. (2005). Competition interfaces and second class particles. Ann. Probab. 33 1235-1254. · Zbl 1078.60083 · doi:10.1214/009117905000000080 · arxiv:math/0406333 [12] Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 79-109. · doi:10.1007/BF00343738 [13] Howard, C. D. and Newman, C. M. (2001). Geodesics and spanning trees for Euclidean first-passage percolation. Ann. Probab. 29 577-623. · Zbl 1062.60099 [14] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437-476. · Zbl 0969.15008 · doi:10.1007/s002200050027 · arxiv:math/9903134 [15] Johansson, K. (2003). Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 277-329. · Zbl 1031.60084 · doi:10.1007/s00220-003-0945-y · arxiv:math/0206208 [16] Kardar, M. and Zhang, Y. C. (1987). Scaling of directed polymers in random media. Phys. Rev. Lett. 58 2087-2090. [17] Moreno Flores, G., Quastel, J. and Remenik, D. (2013). Endpoint distribution of directed polymers in $$1+1$$ dimensions. Comm. Math. Phys. 317 363-380. · Zbl 1257.82117 · doi:10.1007/s00220-012-1583-z · arxiv:1106.2716 [18] Newman, C. M. (1995). A surface view of first-passage percolation. In Proceedings of the International Congress of Mathematicians , Vol. 1, 2 ( Zürich , 1994) 1017-1023. Birkhäuser, Basel. · Zbl 0848.60089 [19] Pimentel, L. P. R. (2014). On the location of the maximum of a continuous stochastic process. J. Appl. Probab. 51 152-161. · Zbl 1305.60029 · doi:10.1239/jap/1395771420 · euclid:jap/1395771420 [20] Quastel, J. and Remenik, D. (2014). Airy processes and variational problems. In Topics in Percolative and Disordered Systems. Springer Proc. Math. Stat. 69 121-171. Springer, New York. · Zbl 1329.82059 · doi:10.1007/978-1-4939-0339-9_5 [21] Rost, H. (1981). Nonequilibrium behaviour of a many particle process: Density profile and local equilibria. Z. Wahrsch. Verw. Gebiete 58 41-53. · Zbl 0451.60097 · doi:10.1007/BF00536194 [22] Wüthrich, M. V. (2002). Asymptotic behaviour of semi-infinite geodesics for maximal increasing subsequences in the plane. In In and Out of Equilibrium ( Mambucaba , 2000). Progress in Probability 51 205-226. Birkhäuser, Boston, MA.
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