The Annals of Probability

Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

Bertrand Duplantier, Rémi Rhodes, Scott Sheffield, and Vincent Vargas

Full-text: Open access

Abstract

In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.

Article information

Source
Ann. Probab., Volume 42, Number 5 (2014), 1769-1808.

Dates
First available in Project Euclid: 29 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1409319467

Digital Object Identifier
doi:10.1214/13-AOP890

Mathematical Reviews number (MathSciNet)
MR3262492

Zentralblatt MATH identifier
1306.60055

Subjects
Primary: 60G57: Random measures 60G15: Gaussian processes 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Gaussian multiplicative chaos Liouville quantum gravity maximum of log-correlated fields

Citation

Duplantier, Bertrand; Rhodes, Rémi; Sheffield, Scott; Vargas, Vincent. Critical Gaussian multiplicative chaos: Convergence of the derivative martingale. Ann. Probab. 42 (2014), no. 5, 1769--1808. doi:10.1214/13-AOP890. https://projecteuclid.org/euclid.aop/1409319467


Export citation

References

  • [1] Aïdékon, E. (2013). Convergence in law of the minimum of a branching random walk. Ann. Probab. 41 1362–1426.
  • [2] Aïdékon, E. and Shi, Z. The Seneta–Heyde scaling for the branching random walk. Available at arXiv:1102.0217v2.
  • [3] Allez, R., Rhodes, R. and Vargas, V. (2013). Lognormal $\star$-scale invariant random measures. Probab. Theory Related Fields 155 751–788.
  • [4] Alvarez-Gaumé, L., Barbón, J. L. F. and Crnković, Č. (1993). A proposal for strings at $D>1$. Nuclear Phys. B 394 383–422.
  • [5] Ambjørn, J., Durhuus, B. and Jónsson, T. (1994). A solvable 2D gravity model with $\gamma>0$. Modern Phys. Lett. A 9 1221–1228.
  • [6] Arguin, L.-P. and Zindy, O. (2014). Poisson–Dirichlet statistics for the extremes of a log-correlated Gaussian field. Ann. Appl. Probab. 24 1446–1481.
  • [7] Bacry, E. and Muzy, J. F. (2003). Log-infinitely divisible multifractal processes. Comm. Math. Phys. 236 449–475.
  • [8] Barral, J. (1999). Moments, continuité, et analyse multifractale des martingales de Mandelbrot. Probab. Theory Related Fields 113 535–569.
  • [9] Barral, J., Jin, X., Rhodes, R. and Vargas, V. (2013). Gaussian multiplicative chaos and KPZ duality. Comm. Math. Phys. 323 451–485.
  • [10] Barral, J., Kupiainen, A., Nikula, M., Saksman, E. and Webb, C. (2014). Critical Mandelbrot cascades. Comm. Math. Phys. 325 685–711.
  • [11] Barral, J. and Mandelbrot, B. B. (2002). Multifractal products of cylindrical pulses. Probab. Theory Related Fields 124 409–430.
  • [12] Barral, J., Rhodes, R. and Vargas, V. (2012). Limiting laws of supercritical branching random walks. C. R. Math. Acad. Sci. Paris 350 535–538.
  • [13] Benjamini, I. and Schramm, O. (2009). KPZ in one dimensional random geometry of multiplicative cascades. Comm. Math. Phys. 289 653–662.
  • [14] Bernardi, O. and Bousquet-Mélou, M. (2011). Counting colored planar maps: Algebraicity results. J. Combin. Theory Ser. B 101 315–377.
  • [15] Biggins, J. D. and Kyprianou, A. E. (2004). Measure change in multitype branching. Adv. in Appl. Probab. 36 544–581.
  • [16] Biggins, J. D. and Kyprianou, A. E. (2005). Fixed points of the smoothing transform: The boundary case. Electron. J. Probab. 10 609–631.
  • [17] Bramson, M. and Zeitouni, O. (2012). Tightness of the recentered maximum of the two-dimensional discrete Gaussian free field. Comm. Pure Appl. Math. 65 1–20.
  • [18] Brézin, É., Kazakov, V. A. and Zamolodchikov, Al. B. (1990). Scaling violation in a field theory of closed strings in one physical dimension. Nuclear Phys. B 338 673–688.
  • [19] Carpentier, D. and Le Doussal, P. (2001). Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and Sinh–Gordon models. Phys. Rev. E (3) 63 026110.
  • [20] Daley, D. J. and Vere-Jones, D. (2007). An Introduction to the Theory of Point Processes. Volume 2: Probability and Its Applications, 2nd ed. Springer, New York.
  • [21] Das, S. R., Dhar, A., Sengupta, A. M. and Wadia, S. R. (1990). New critical behavior in $d=0$ large-$N$ matrix models. Modern Phys. Lett. A 5 1041–1056.
  • [22] Daul, J.-M. $q$-state Potts model on a random planar lattice. Available at arXiv:hep-th/9502014.
  • [23] David, F. (1988). Conformal field theories coupled to 2-D gravity in the conformal gauge. Modern Phys. Lett. A 3 1651–1656.
  • [24] Ding, J. and Zeitouni, O. Extreme values for two-dimensional discrete Gaussian free field. Available at arXiv:1206.0346v1.
  • [25] Distler, J. and Kawai, H. (1989). Conformal field theory and 2-D quantum gravity. Nuclear Phys. B 321 509–527.
  • [26] Di Francesco, P., Ginsparg, P. and Zinn-Justin, J. (1995). 2D gravity and random matrices. Phys. Rep. 254 1–133.
  • [27] Duplantier, B. (2004). Conformal fractal geometry and boundary quantum gravity. In Fractal Geometry and Applications: A Jubilee of BenoîT Mandelbrot, Part 2. Proc. Sympos. Pure Math. 72 365–482. Amer. Math. Soc., Providence, RI.
  • [28] Duplantier, B. (2010). A rigorous perspective on Liouville quantum gravity and KPZ. In Exact Methods in Low-Dimensional Statistical Physics and Quantum Computing (J. Jacobsen, S. Ouvry, V. Pasquier, D. Serban and L. F. Cugliandolo, eds.) 529–561. Oxford Univ. Press, Oxford.
  • [29] Duplantier, B. and Sheffield, S. (2009). Duality and the Knizhnik–Polyakov–Zamolodchikov relation in Liouville quantum gravity. Phys. Rev. Lett. 102 150603.
  • [30] Duplantier, B. and Sheffield, S. (2011). Liouville quantum gravity and KPZ. Invent. Math. 185 333–393.
  • [31] Duplantier, B. and Sheffield, S. (2011). Schramm–Loewner evolution and Liouville quantum gravity. Phys. Rev. Lett. 107 131305.
  • [32] Durhuus, B. (1994). Multi-spin systems on a randomly triangulated surface. Nuclear Phys. B 426 203–222.
  • [33] Durrett, R. and Liggett, T. M. (1983). Fixed points of the smoothing transformation. Probab. Theory Related Fields 64 275–301.
  • [34] Eynard, B. and Bonnet, G. (1999). The Potts-$q$ random matrix model: Loop equations, critical exponents, and rational case. Phys. Lett. B 463 273–279.
  • [35] Fan, A. H. (1997). Sur les chaos de Lévy stables d’indice $0<\alpha<1$. Ann. Sci. Math. Qué. 21 53–66.
  • [36] Fyodorov, Y. V. and Bouchaud, J.-P. (2008). Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential. J. Phys. A 41 372001.
  • [37] Fyodorov, Y. V., Le Doussal, P. and Rosso, A. (2009). Statistical mechanics of logarithmic REM: Duality, freezing and extreme value statistics of $1/f$ noises generated by Gaussian free fields J. Stat. Mech. P10005.
  • [38] Ginsparg, P. and Moore, G. (1993). Lectures on 2D gravity and 2D string theory. In Recent Direction in Particle Theory (J. Harvey and J. Polchinski, eds.). World Scientific, Singapore.
  • [39] Ginsparg, P. and Zinn-Justin, J. (1990). 2D gravity +1D matter. Phys. Lett. B 240 333–340.
  • [40] Gross, D. J. and Klebanov, I. (1990). One-dimensional string theory on a circle. Nuclear Phys. B 344 475–498.
  • [41] Gross, D. J. and Miljković, N. (1990). A nonperturbative solution of $D=1$ string theory. Phys. Lett. B 238 217–223.
  • [42] Gubser, S. S. and Klebanov, I. R. (1994). A modified $c=1$ matrix model with new critical behavior. Phys. Lett. B 340 35–42.
  • [43] 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.
  • [44] Jain, S. and Mathur, S. D. (1992). World-sheet geometry and baby universes in 2D quantum gravity. Phys. Lett. B 286 239–246.
  • [45] Kahane, J.-P. (1985). Sur le chaos multiplicatif. Ann. Sci. Math. Qué. 9 105–150.
  • [46] Kahane, J.-P. and Peyrière, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22 131–145.
  • [47] Kazakov, V., Kostov, I. and Kutasov, D. (2000). A matrix model for the 2d black hole. In Nonperturbative Quantum Effects. JHEP Proceedings. Available at http://pos.sissa.it/cgi-bin/reader/conf.cgi?confid=6.
  • [48] Klebanov, I. R. (1995). Touching random surfaces and Liouville gravity. Phys. Rev. D 51 1836–1841.
  • [49] Klebanov, I. R. and Hashimoto, A. (1995). Non-perturbative solution of matrix models modified by trace-squared terms. Nuclear Phys. B 434 264–282.
  • [50] Klebanov, I. R. and Hashimoto, A. (1996). Wormholes, matrix models, and Liouville gravity. Nuclear Phys. B Proc. Suppl. 45BC 135–148. String theory, gauge theory and quantum gravity (Trieste, 1995).
  • [51] Knizhnik, V. G., Polyakov, A. M. and Zamolodchikov, A. B. (1988). Fractal structure of 2D-quantum gravity. Modern Phys. Lett. A 3 819–826.
  • [52] Kostov, I. K. (1991). Loop amplitudes for nonrational string theories. Phys. Lett. B 266 317–324.
  • [53] Kostov, I. K. (1992). Strings with discrete target space. Nuclear Phys. B 376 539–598.
  • [54] Kostov, I. K. (2010). Boundary loop models and 2D quantum gravity. In Exact Methods in Low-Dimensional Statistical Physics and Quantum Computing (J. Jacobsen, S. Ouvry, V. Pasquier, D. Serban and L. F. Cugliandolo, eds.) 363–406. Oxford Univ. Press, Oxford.
  • [55] Kostov, I. K. and Staudacher, M. (1992). Multicritical phases of the $O(n)$ model on a random lattice. Nuclear Phys. B 384 459–483.
  • [56] Kyprianou, A. E. (1998). Slow variation and uniqueness of solutions to the functional equation in the branching random walk. J. Appl. Probab. 35 795–801.
  • [57] Liu, Q. (1998). Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. in Appl. Probab. 30 85–112.
  • [58] Madaule, T. Convergence in law for the branching random walk seen from its tip. Available at arXiv:1107.2543v2.
  • [59] Mandelbrot, B. B. (1972). A possible refinement of the lognormal hypothesis concerning the distribution of energy in intermittent turbulence. In Statistical Models and Turbulence 333–351. Springer, New York.
  • [60] Motoo, M. (1958). Proof of the law of iterated logarithm through diffusion equation. Ann. Inst. Statist. Math. 10 21–28.
  • [61] Nakayama, Y. (2004). Liouville field theory: A decade after the revolution. Internat. J. Modern Phys. A 19 2771–2930.
  • [62] Neveu, J. (1988). Multiplicative martingales for spatial branching processes. In Seminar on Stochastic Processes, 1987 (Princeton, NJ, 1987) (E. Cinlar, K. L. Chung and R. K. Getoor, eds.). Progr. Probab. Statist. 15 223–241. Birkhäuser, Boston, MA.
  • [63] Nienhuis, B. (1987). Coulomb gas formulation of two-dimensional phase transitions. In Phase Transitions and Critical Phenomena (C. Domb and J. L. Lebowitz, eds.). Academic Press, London.
  • [64] Parisi, G. (1990). On the one-dimensional discretized string. Phys. Lett. B 238 209–212.
  • [65] Polchinski, J. (1990). Critical behavior of random surfaces in one dimension. Nuclear Phys. B 346 253–263.
  • [66] Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Probab. Theory Related Fields 82 451–487.
  • [67] Rhodes, R., Sohier, J. and Vargas, V. (2014). Levy multiplicative chaos and star scale invariant random measures. Ann. Probab. 42 689–724.
  • [68] Rhodes, R. and Vargas, V. (2010). Multidimensional multifractal random measures. Electron. J. Probab. 15 241–258.
  • [69] Rhodes, R. and Vargas, V. (2011). KPZ formula for log-infinitely divisible multifractal random measures. ESAIM Probab. Stat. 15 358–371.
  • [70] Sheffield, S. Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Available at arXiv:1012.4797.
  • [71] Sheffield, S. (2007). Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 521–541.
  • [72] Sugino, F. and Tsuchiya, O. (1994). Critical behavior in $c=1$ matrix model with branching interactions. Modern Phys. Lett. A 9 3149–3162.
  • [73] Tecu, N. Random conformal welding at criticality. Available at arXiv:1205.3189v1.