• Bernoulli
  • Volume 24, Number 2 (2018), 801-841.

The minimum of a branching random walk outside the boundary case

Julien Barral, Yueyun Hu, and Thomas Madaule

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This paper is a complement to the studies on the minimum of a real-valued branching random walk. In the boundary case [Electron. J. Probab. 10 (2005) 609–631], Aïdékon in a seminal paper [Ann. Probab. 41 (2013) 1362–1426] obtained the convergence in law of the minimum after a suitable renormalization. We study here the situation when the log-generating function of the branching random walk explodes at some positive point and it cannot be reduced to the boundary case. In the associated thermodynamics framework, this corresponds to a first-order phase transition, while the boundary case corresponds to a second-order phase transition.

Article information

Bernoulli, Volume 24, Number 2 (2018), 801-841.

Received: October 2014
Revised: October 2015
First available in Project Euclid: 21 September 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

branching random walk minimal position phase transition


Barral, Julien; Hu, Yueyun; Madaule, Thomas. The minimum of a branching random walk outside the boundary case. Bernoulli 24 (2018), no. 2, 801--841. doi:10.3150/15-BEJ784.

Export citation


  • [1] Addario-Berry, L. and Reed, B. (2009). Minima in branching random walks. Ann. Probab. 37 1044–1079.
  • [2] Aïdékon, E. (2013). Convergence in law of the minimum of a branching random walk. Ann. Probab. 41 1362–1426.
  • [3] Alsmeyer, G., Biggins, J.D. and Meiners, M. (2012). The functional equation of the smoothing transform. Ann. Probab. 40 2069–2105.
  • [4] Attia, N. and Barral, J. (2014). Hausdorff and packing spectra, large deviations, and free energy for branching random walks in $\mathbb{R}^{d}$. Comm. Math. Phys. 331 139–187.
  • [5] Bacry, E. and Muzy, J.F. (2003). Log-infinitely divisible multifractal processes. Comm. Math. Phys. 236 449–475.
  • [6] Barral, J., Kupiainen, A., Nikula, M., Saksman, E. and Webb, C. (2014). Critical Mandelbrot cascades. Comm. Math. Phys. 325 685–711.
  • [7] Barral, J. and Mandelbrot, B.B. (2002). Multifractal products of cylindrical pulses. Probab. Theory Related Fields 124 409–430.
  • [8] Barral, J., Rhodes, R. and Vargas, V. (2012). Limiting laws of supercritical branching random walks. C. R. Math. Acad. Sci. Paris 350 535–538.
  • [9] Biggins, J.D. (1976). The first- and last-birth problems for a multitype age-dependent branching process. Adv. in Appl. Probab. 8 446–459.
  • [10] Biggins, J.D. (1977). Martingale convergence in the branching random walk. J. Appl. Probab. 14 25–37.
  • [11] Biggins, J.D. (1992). Uniform convergence of martingales in the branching random walk. Ann. Probab. 20 137–151.
  • [12] Biggins, J.D. and Kyprianou, A.E. (1997). Seneta–Heyde norming in the branching random walk. Ann. Probab. 25 337–360.
  • [13] Biggins, J.D. and Kyprianou, A.E. (2004). Measure change in multitype branching. Adv. in Appl. Probab. 36 544–581.
  • [14] Biggins, J.D. and Kyprianou, A.E. (2005). Fixed points of the smoothing transform: The boundary case. Electron. J. Probab. 10 609–631.
  • [15] Bramson, M., Ding, J. and Zeitouni, O. (2013). Convergence in law of the maximum of the two-dimensional discrete Gaussian free field. Preprint. Available at
  • [16] Bramson, M., Ding, J. and Zeitouni, O. (2014). Convergence in law of the maximum of nonlattice branching random walk. Preprint. Available at
  • [17] Bramson, M. and Zeitouni, O. (2009). Tightness for a family of recursion equations. Ann. Probab. 37 615–653.
  • [18] Chauvin, B., Rouault, A. and Wakolbinger, A. (1991). Growing conditioned trees. Stochastic Process. Appl. 39 117–130.
  • [19] Chen, X. (2015). A necessary and sufficient condition for the nontrivial limit of the derivative martingale in a branching random walk. Adv. in Appl. Probab. 47 741–760.
  • [20] Collet, P. and Koukiou, F. (1992). Large deviations for multiplicative chaos. Comm. Math. Phys. 147 329–342.
  • [21] Denisov, D., Dieker, A.B. and Shneer, V. (2008). Large deviations for random walks under subexponentiality: The big-jump domain. Ann. Probab. 36 1946–1991.
  • [22] Derrida, B. and Spohn, H. (1988). Polymers on disordered trees, spin glasses, and traveling waves. J. Stat. Phys. 51 817–840.
  • [23] Ding, J., Roy, R. and Zeitouni, O. (2015). Convergence of the centered maximum of log-correlated Gaussian fields. Preprint. Available at
  • [24] Durrett, R. and Liggett, T.M. (1983). Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete 64 275–301.
  • [25] Fan, A.H. (1997). Sur les chaos de Lévy stables d’indice $0<\alpha<1$. Ann. Sci. Math. Québec 21 53–66.
  • [26] Gut, A. (2009). Stopped Random Walks: Limit Theorems and Applications, 2nd ed. Springer Series in Operations Research and Financial Engineering. New York: Springer.
  • [27] Hammersley, J.M. (1974). Postulates for subadditive processes. Ann. Probab. 2 652–680.
  • [28] Hu, Y. and Shi, Z. (2009). Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37 742–789.
  • [29] Kahane, J.-P. and Peyrière, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Adv. in Math. 22 131–145.
  • [30] Kingman, J.F.C. (1975). The first birth problem for an age-dependent branching process. Ann. Probab. 3 790–801.
  • [31] Kyprianou, A.E. (2000). Martingale convergence and the stopped branching random walk. Probab. Theory Related Fields 116 405–419.
  • [32] Liu, Q. (1998). Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. in Appl. Probab. 30 85–112.
  • [33] Lyons, R. (1997). A simple path to Biggins’ martingale convergence for branching random walk. In Classical and Modern Branching Processes (Minneapolis, MN, 1994). IMA Vol. Math. Appl. 84 217–221. New York: Springer.
  • [34] Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes. Ann. Probab. 23 1125–1138.
  • [35] Madaule, T. (2011). Convergence in law for the branching random walk seen from its tip. Preprint. Available at
  • [36] Madaule, T. (2015). Maximum of a log-correlated Gaussian field. Ann. Inst. Henri Poincaré Probab. Stat. 51 1369–1431.
  • [37] Madaule, T., Rhodes, R. and Vargas, V. (2016). Glassy phase and freezing of log-correlated Gaussian potentials. Ann. Appl. Probab. To appear. Available at arXiv:1310.5574.
  • [38] Mandelbrot, B. (1974). Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire. C. R. Math. Acad. Sci. Paris 278 289–292.
  • [39] Molchan, G.M. (1996). Scaling exponents and multifractal dimensions for independent random cascades. Comm. Math. Phys. 179 681–702.
  • [40] Mörters, P. and Ortgiese, M. (2008). Minimal supporting subtrees for the free energy of polymers on disordered trees. J. Math. Phys. 49 125203, 21.
  • [41] Ossiander, M. and Waymire, E.C. (2000). Statistical estimation for multiplicative cascades. Ann. Statist. 28 1533–1560.
  • [42] Rhodes, R., Sohier, J. and Vargas, V. (2014). Levy multiplicative chaos and star scale invariant random measures. Ann. Probab. 42 689–724.
  • [43] Shi, Z. (2016). Branching Random Walks. École D’été de Saint-Flour XLII. To appear.
  • [44] Stone, C. (1965). A local limit theorem for nonlattice multi-dimensional distribution functions. Ann. Math. Stat. 36 546–551.
  • [45] Vatutin, V.A. and Wachtel, V. (2009). Local probabilities for random walks conditioned to stay positive. Probab. Theory Related Fields 143 177–217.
  • [46] Waymire, E.C. and Williams, S.C. (1996). A cascade decomposition theory with applications to Markov and exchangeable cascades. Trans. Amer. Math. Soc. 348 585–632.
  • [47] Webb, C. (2011). Exact asymptotics of the freezing transition of a logarithmically correlated random energy model. J. Stat. Phys. 145 1595–1619.