## The Annals of Probability

### The range of tree-indexed random walk in low dimensions

#### Abstract

We study the range $R_{n}$ of a random walk on the $d$-dimensional lattice $\mathbb{Z}^{d}$ indexed by a random tree with $n$ vertices. Under the assumption that the random walk is centered and has finite fourth moments, we prove in dimension $d\leq3$ that $n^{-d/4}R_{n}$ converges in distribution to the Lebesgue measure of the support of the integrated super-Brownian excursion (ISE). An auxiliary result shows that the suitably rescaled local times of the tree-indexed random walk converge in distribution to the density process of ISE. We obtain similar results for the range of critical branching random walk in $\mathbb{Z}^{d}$, $d\leq3$. As an intermediate estimate, we get exact asymptotics for the probability that a critical branching random walk starting with a single particle at the origin hits a distant point. The results of the present article complement those derived in higher dimensions in our earlier work.

#### Article information

Source
Ann. Probab., Volume 43, Number 5 (2015), 2701-2728.

Dates
First available in Project Euclid: 9 September 2015

https://projecteuclid.org/euclid.aop/1441792296

Digital Object Identifier
doi:10.1214/14-AOP947

Mathematical Reviews number (MathSciNet)
MR3395472

Zentralblatt MATH identifier
1335.60070

#### Citation

Le Gall, Jean-François; Lin, Shen. The range of tree-indexed random walk in low dimensions. Ann. Probab. 43 (2015), no. 5, 2701--2728. doi:10.1214/14-AOP947. https://projecteuclid.org/euclid.aop/1441792296

#### References

• [1] Aldous, D. (1993). Tree-based models for random distribution of mass. J. Stat. Phys. 73 625–641.
• [2] Benjamini, I. and Curien, N. (2012). Recurrence of the $\mathbb{Z}^{d}$-valued infinite snake via unimodularity. Electron. Commun. Probab. 17 10.
• [3] Bousquet-Mélou, M. and Janson, S. (2006). The density of the ISE and local limit laws for embedded trees. Ann. Appl. Probab. 16 1597–1632.
• [4] Dawson, D. A., Iscoe, I. and Perkins, E. A. (1989). Super-Brownian motion: Path properties and hitting probabilities. Probab. Theory Related Fields 83 135–205.
• [5] Devroye, L. and Janson, S. (2011). Distances between pairs of vertices and vertical profile in conditioned Galton–Watson trees. Random Structures Algorithms 38 381–395.
• [6] Janson, S. and Marckert, J.-F. (2005). Convergence of discrete snakes. J. Theoret. Probab. 18 615–647.
• [7] Lalley, S. P. and Zheng, X. (2010). Spatial epidemics and local times for critical branching random walks in dimensions 2 and 3. Probab. Theory Related Fields 148 527–566.
• [8] Lalley, S. P. and Zheng, X. (2011). Occupation statistics of critical branching random walks in two or higher dimensions. Ann. Probab. 39 327–368.
• [9] Lawler, G. F. and Limic, V. (2010). Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics 123. Cambridge Univ. Press, Cambridge.
• [10] Le Gall, J.-F. (1995). The Brownian snake and solutions of $\Delta u=u^{2}$ in a domain. Probab. Theory Related Fields 102 393–432.
• [11] Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser, Basel.
• [12] Le Gall, J.-F. (2005). Random trees and applications. Probab. Surv. 2 245–311.
• [13] Le Gall, J.-F. and Lin, S. (2014). The range of tree-indexed random walk. J. Inst. Math. Jussieu. To appear. DOI:10/1017/S1474748014000280.
• [14] Le Gall, J.-F. and Miermont, G. (2012). Scaling limits of random trees and planar maps. In Probability and Statistical Physics in Two and More Dimensions. Clay Math. Proc. 15 155–211. Amer. Math. Soc., Providence, RI.
• [15] Le Gall, J.-F. and Weill, M. (2006). Conditioned Brownian trees. Ann. Inst. Henri Poincaré Probab. Stat. 42 455–489.
• [16] Marckert, J.-F. and Mokkadem, A. (2003). The depth first processes of Galton–Watson trees converge to the same Brownian excursion. Ann. Probab. 31 1655–1678.
• [17] Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Springer, Berlin.
• [18] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin.
• [19] Sugitani, S. (1989). Some properties for the measure-valued branching diffusion processes. J. Math. Soc. Japan 41 437–462.