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Critical Gaussian multiplicative chaos: convergence of the derivative martingale. (English) Zbl 1306.60055
Summary: 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 atoms. In connection with the derivative martingale, we state 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.

60G57 Random measures
60G15 Gaussian processes
60D05 Geometric probability and stochastic geometry
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[1] Aïdékon, E. (2013). Convergence in law of the minimum of a branching random walk. Ann. Probab. 41 1362-1426. · Zbl 1285.60086
[2] Aïdékon, E. and Shi, Z. The Seneta-Heyde scaling for the branching random walk. Available at . 1102.0217v2
[3] Allez, R., Rhodes, R. and Vargas, V. (2013). Lognormal \(\star\)-scale invariant random measures. Probab. Theory Related Fields 155 751-788. · Zbl 1278.60083
[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. · Zbl 1022.81722
[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. · Zbl 1301.60042
[7] Bacry, E. and Muzy, J. F. (2003). Log-infinitely divisible multifractal processes. Comm. Math. Phys. 236 449-475. · Zbl 1032.60046
[8] Barral, J. (1999). Moments, continuité, et analyse multifractale des martingales de Mandelbrot. Probab. Theory Related Fields 113 535-569. · Zbl 0936.60045
[9] Barral, J., Jin, X., Rhodes, R. and Vargas, V. (2013). Gaussian multiplicative chaos and KPZ duality. Comm. Math. Phys. 323 451-485. · Zbl 1287.83019
[10] Barral, J., Kupiainen, A., Nikula, M., Saksman, E. and Webb, C. (2014). Critical Mandelbrot cascades. Comm. Math. Phys. 325 685-711. · Zbl 1302.60065
[11] Barral, J. and Mandelbrot, B. B. (2002). Multifractal products of cylindrical pulses. Probab. Theory Related Fields 124 409-430. · Zbl 1014.60042
[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. · Zbl 1260.60173
[13] Benjamini, I. and Schramm, O. (2009). KPZ in one dimensional random geometry of multiplicative cascades. Comm. Math. Phys. 289 653-662. · Zbl 1170.83006
[14] Bernardi, O. and Bousquet-Mélou, M. (2011). Counting colored planar maps: Algebraicity results. J. Combin. Theory Ser. B 101 315-377. · Zbl 1223.05123
[15] Biggins, J. D. and Kyprianou, A. E. (2004). Measure change in multitype branching. Adv. in Appl. Probab. 36 544-581. · Zbl 1056.60082
[16] Biggins, J. D. and Kyprianou, A. E. (2005). Fixed points of the smoothing transform: The boundary case. Electron. J. Probab. 10 609-631. · Zbl 1110.60081
[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. · Zbl 1237.60041
[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. · Zbl 0657.60069
[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. · Zbl 1020.81740
[22] Daul, J.-M. \(q\)-state Potts model on a random planar lattice. Available at .
[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 . 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. · Zbl 1068.60019
[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. · Zbl 1208.83040
[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. · Zbl 1226.81241
[31] Duplantier, B. and Sheffield, S. (2011). Schramm-Loewner evolution and Liouville quantum gravity. Phys. Rev. Lett. 107 131305. · Zbl 1226.81241
[32] Durhuus, B. (1994). Multi-spin systems on a randomly triangulated surface. Nuclear Phys. B 426 203-222. · Zbl 1049.82521
[33] Durrett, R. and Liggett, T. M. (1983). Fixed points of the smoothing transformation. Probab. Theory Related Fields 64 275-301. · Zbl 0506.60097
[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. · Zbl 1037.82522
[35] Fan, A. H. (1997). Sur les chaos de Lévy stables d’indice \(0<\alpha<1\). Ann. Sci. Math. Qué. 21 53-66. · Zbl 0884.60040
[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. · Zbl 1214.82016
[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. · Zbl 1332.81175
[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. · Zbl 1169.60021
[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. · Zbl 0596.60041
[46] Kahane, J.-P. and Peyrière, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22 131-145. · Zbl 0349.60051
[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 . · Zbl 0988.81099
[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. · Zbl 1020.81751
[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). · Zbl 0991.81582
[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. · Zbl 1219.83003
[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. · Zbl 0930.60066
[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. · Zbl 0909.60075
[58] Madaule, T. Convergence in law for the branching random walk seen from its tip. Available at . 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. · Zbl 0227.76081
[60] Motoo, M. (1958). Proof of the law of iterated logarithm through diffusion equation. Ann. Inst. Statist. Math. 10 21-28. · Zbl 0084.35801
[61] Nakayama, Y. (2004). Liouville field theory: A decade after the revolution. Internat. J. Modern Phys. A 19 2771-2930. · Zbl 1080.81056
[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. · Zbl 0652.60089
[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. · Zbl 1332.81185
[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. · Zbl 0659.60078
[67] Rhodes, R., Sohier, J. and Vargas, V. (2014). Levy multiplicative chaos and star scale invariant random measures. Ann. Probab. 42 689-724. · Zbl 1295.60064
[68] Rhodes, R. and Vargas, V. (2010). Multidimensional multifractal random measures. Electron. J. Probab. 15 241-258. · Zbl 1201.60046
[69] Rhodes, R. and Vargas, V. (2011). KPZ formula for log-infinitely divisible multifractal random measures. ESAIM Probab. Stat. 15 358-371. · Zbl 1268.60070
[70] Sheffield, S. Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Available at . 1012.4797 · Zbl 1388.60144
[71] Sheffield, S. (2007). Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 521-541. · Zbl 1132.60072
[72] Sugino, F. and Tsuchiya, O. (1994). Critical behavior in \(c=1\) matrix model with branching interactions. Modern Phys. Lett. A 9 3149-3162. · Zbl 1015.81572
[73] Tecu, N. Random conformal welding at criticality. Available at . 1205.3189v1
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