Electronic Journal of Probability

The phase diagram of the complex branching Brownian motion energy model

Lisa Hartung and Anton Klimovsky

Full-text: Open access


Branching Brownian motion (BBM) is a convenient representative of the class of log-correlated random fields. Motivated by the conjectured criticality of the log-correlated fields, we take the viewpoint of statistical physics on the BBM: We consider the partition function of the field of energies given by the “positions” of the particles of the complex-valued BBM. In such a complex BBM energy model, we allow for arbitrary correlations between the real and imaginary parts of the energies. We identify the fluctuations of the partition function. As a consequence, we get the full phase diagram of the log-partition function. It turns out that the phase diagram is the same as for the field of independent energies, i.e., Derrida’s random energy model (REM). Yet, the fluctuations are different from those of the REM in all phases. All results are shown for any correlation between the real and imaginary parts of the random energy.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 127, 27 pp.

Received: 18 April 2017
Accepted: 16 November 2018
First available in Project Euclid: 19 December 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G70: Extreme value theory; extremal processes 60F05: Central limit and other weak theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Gaussian processes branching Brownian motion logarithmic correlations random energy model phase diagram central limit theorem random variance martingale convergence

Creative Commons Attribution 4.0 International License.


Hartung, Lisa; Klimovsky, Anton. The phase diagram of the complex branching Brownian motion energy model. Electron. J. Probab. 23 (2018), paper no. 127, 27 pp. doi:10.1214/18-EJP245. https://projecteuclid.org/euclid.ejp/1545188694

Export citation


  • [1] E. Aïdékon and Z. Shi. The Seneta-Heyde scaling for the branching random walk. Ann. Probab., 42(3):959–993, 2014.
  • [2] G. Alsmeyer and M. Meiners. Fixed points of the smoothing transform: two-sided solutions. Probab. Theory Relat. Fields, 155(1-2):165–199, 2013.
  • [3] L.-P. Arguin, D. Belius, and P. Bourgade. Maximum of the characteristic polynomial of random unitary matrices. Comm. Math. Phys., 349(2):703–751, 2017.
  • [4] L.-P. Arguin, D. Belius, and A.J. Harper. Maxima of a randomized Riemann Zeta function, and branching random walks. Ann. Appl. Probab., 27(1):178–215, 2017.
  • [5] K.B. Athreya and P.E. Ney. Branching processes. Springer-Verlag, New York-Heidelberg, 1972.
  • [6] J. Barral, X. Jin, and B. Mandelbrot. Convergence of complex multiplicative cascades. Ann. Appl. Probab., 20(4):1219–1252, 2010.
  • [7] J. Barral, X. Jin, and B. Mandelbrot. Uniform convergence for complex $[0,1]$-martingales. Ann. Appl. Probab., 20(4):1205–1218, 2010.
  • [8] A. Bovier. Gaussian Processes on Trees: From Spin-Glasses to Branching Brownian Motion. Cambridge University Press, 2016.
  • [9] A. Bovier and L. Hartung. The extremal process of two-speed branching Brownian motion. Electron. J. Probab., 19(18):1–28, 2014.
  • [10] A. Bovier and I. Kurkova. Derrida’s generalised random energy models. I. Models with finitely many hierarchies. Ann. Inst. H. Poincaré Probab. Statist., 40(4):439–480, 2004.
  • [11] A. Bovier and I. Kurkova. Derrida’s generalized random energy models. II. Models with continuous hierarchies. Ann. Inst. H. Poincaré Probab. Statist., 40(4):481–495, 2004.
  • [12] B. Derrida. Random-Energy Model: Limit of a Family of Disordered Models. Phys. Rev. Lett., 45:79–82, 1980.
  • [13] B. Derrida. A generalization of the Random Energy Model which includes correlations between energies. J. Physique Lett., 46(9):401–407, 1985.
  • [14] B. Derrida. The zeroes of the partition function of the random energy model. Physica A: Stat. Mech. Appl., 177:31–37, 1991.
  • [15] B. Derrida, M.R. Evans, and E.R. Speer. Mean field theory of directed polymers with random complex weights. Comm. Math. Phys., 156(2):221–244, 1993.
  • [16] A. Dobrinevski, P. Le Doussal, and K.J. Wiese. Interference in disordered systems: A particle in a complex random landscape. Phys. Rev. E, 83(6):061116, 2011.
  • [17] Y.V. Fyodorov, G.A. Hiary, and J.P. Keating. Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function. Phys. Rev. Lett., 108(17):170601, 2012.
  • [18] M. Hairer and H. Shen. The dynamical sine-Gordon model. Comm. Math. Phys., 341(3):933–989, 2016.
  • [19] L. Hartung and A. Klimovsky. The glassy phase of the complex branching Brownian motion energy model. Electron. Commun. Probab., 20, 2015.
  • [20] A. Iksanov and Z. Kabluchko. A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk. J. Appl. Probab., 53(4):1178–1192, 2016.
  • [21] A. Iksanov, K. Kolesko, and M. Meiners. Fluctuations of Biggins’ martingales at complex parameters. Preprint, 2018. Available at https://arxiv.org/abs/1806.09943.
  • [22] A. Iksanov and M. Meiners. Fixed points of multivariate smoothing transforms with scalar weights. ALEA Lat. Am. J. Probab. Math. Stat., 12(1):69–114, 2015.
  • [23] J. Junnila, E. Saksman, and C. Webb. Imaginary multiplicative chaos: Moments, regularity and connections to the Ising model. Preprint, 2018. Available at https://arxiv.org/abs/1806.02118.
  • [24] Z. Kabluchko and A. Klimovsky. Complex random energy model: zeros and fluctuations. Probab. Theory Relat. Fields, 158(1-2):159–196, 2014.
  • [25] Z. Kabluchko and A. Klimovsky. Generalized random energy model at complex temperatures. Preprint, 2014. Available at http://arxiv.org/abs/1402.2142.
  • [26] N. Kistler. Derrida’s random energy models. From spin glasses to the extremes of correlated random fields. In V. Gayrard and N. Kistler, editors, Correlated Random Systems: Five Different Methods. Springer, 2015.
  • [27] K. Kolesko and M. Meiners. Convergence of complex martingales in the branching random walk: The boundary. Electron. Commun. Probab., 22, 2017.
  • [28] A.E. Kyprianou and T. Madaule. The Seneta-Heyde scaling for homogeneous fragmentations. Preprint, 2015. Available at https://arxiv.org/abs/1507.01559.
  • [29] H. Lacoin, R. Rhodes, and V. Vargas. Complex Gaussian multiplicative chaos. Comm. Math. Phys., 337(2):569–632, 2015.
  • [30] S.P. Lalley and T. Sellke. A conditional limit theorem for the frontier of a branching Brownian motion. Ann. Probab., 15(3):1052–1061, 1987.
  • [31] T.D. Lee and C.N. Yang. Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model. Phys. Rev., 87:410–419, 1952.
  • [32] T. Madaule, R. Rhodes, and V. Vargas. Continuity estimates for the complex cascade model on the phase boundary. Preprint, 2015. Available at http://arxiv.org/abs/1502.05655.
  • [33] T. Madaule, R. Rhodes, and V. Vargas. The glassy phase of complex branching Brownian motion. Comm. Math. Phys., 334(3):1157–1187, 2015.
  • [34] P. Maillard and M. Pain. 1-stable fluctuations in branching Brownian motion at critical temperature I: the derivative martingale. Preprint, 2018. Available at https://arxiv.org/abs/1806.05152.
  • [35] M. Meiners and S. Mentemeier. Solutions to complex smoothing equations. Probab. Theory Relat. Fields, pages 1–70, 2016.
  • [36] D. Panchenko. The Sherrington-Kirkpatrick model. 156 pp. Springer, 2013.
  • [37] R. Rhodes and V. Vargas. Gaussian multiplicative chaos and applications: A review. Probab. Surv., 11:315–392, 2014.
  • [38] E. Saksman and C. Webb. The Riemann zeta function and Gaussian multiplicative chaos: statistics on the critical line. Preprint, 2016. Available at https://arxiv.org/abs/1609.00027.