The Dickman–Goncharov distribution

S. A. Molchanov   ; V. A. Panov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 75 (2020) no. 6, pp. 1089-1132

Résumé

In the 1930s and 40s, one and the same delay differential equation appeared in papers by two mathematicians, Karl Dickman and Vasily Leonidovich Goncharov, who dealt with completely different problems. Dickman investigated the limit value of the number of natural numbers free of large prime factors, while Goncharov examined the asymptotics of the maximum cycle length in decompositions of random permutations. The equation obtained in these papers defines, under a certain initial condition, the density of a probability distribution now called the Dickman–Goncharov distribution (this term was first proposed by Vershik in 1986). Recently, a number of completely new applications of the Dickman–Goncharov distribution have appeared in mathematics (random walks on solvable groups, random graph theory, and so on) and also in biology (models of growth and evolution of unicellular populations), finance (theory of extreme phenomena in finance and insurance), physics (the model of random energy levels), and other fields. Despite the extensive scope of applications of this distribution and of more general but related models, all the mathematical aspects of this topic (for example, infinite divisibility and absolute continuity) are little known even to specialists in limit theorems. The present survey is intended to fill this gap. Both known and new results are given. Bibliography: 62 titles.

Détail

Keywords: Dickman–Goncharov distribution, Vershik chain, Erdős problem, random energy model, cell growth model, random walks on solvable groups.

Bibliographie

[1] A. Alhakim, S. Molchanov, “The density flatness phenomenon”, Statist. Probab. Lett., 152 (2019), 156–161 | DOI | MR | Zbl

[2] K. Alladi, “The Turán–Kubilius inequality for integers without large prime factors”, J. Reine Angew. Math., 1982:335 (1982), 180–196 | DOI | MR | Zbl

[3] Z. D. Bai, Sungchul Lee, M. D. Penrose, “Rooted edges of a minimal directed spanning tree on random points”, Adv. in Appl. Probab., 38:1 (2006), 1–30 | DOI | MR | Zbl

[4] G. Ben Arous, L. V. Bogachev, S. A. Molchanov, “Limit theorems for sums of random exponentials”, Probab. Theory Related Fields, 132:4 (2005), 579–612 | DOI | MR | Zbl

[5] A. G. Bhatt, R. Roy, “On a random directed spanning tree”, Adv. in Appl. Probab., 36:1 (2004), 19–42 | DOI | MR | Zbl

[6] P. Billingsley, “On the distribution of large prime divisors”, Period. Math. Hungar., 2:1-4 (1972), 283–289 | DOI | MR | Zbl

[7] A. Bovier, I. Kurkova, M. Löwe, “Fluctuations of the free energy in the REM and the $p$-spin SK models”, Ann. Probab., 30:2 (2002), 605–651 | DOI | MR | Zbl

[8] A. A. Bukhshtab, “O chislakh arifmeticheskoi progressii, u kotorykh vse prostye mnozhiteli maly po poryadku rosta”, Dokl. AN SSSR, 67 (1949), 5–8 | MR | Zbl

[9] S. Chatterjee, Sanchayan Sen, “Minimal spanning trees and Stein's method”, Ann. Appl. Probab., 27:3 (2017), 1588–1645 | DOI | MR | Zbl

[10] R. Cont, P. Tankov, Financial modelling with jump processes, Chapman Hall/CRC Financ. Math. Ser., Chapman Hall/CRC, Boca Raton, FL, 2004, xvi+535 pp. | DOI | MR | Zbl

[11] S. Covo, “On approximations of small jumps of subordinators with particular emphasis on a Dickman-type limit”, J. Appl. Probab., 46:3 (2009), 732–755 | DOI | MR | Zbl

[12] N. G. de Bruijn, “On the number of positive integers $\leq x $ and free of prime factors ${> y}$”, Nederl. Acad. Wetensch. Proc. Ser. A, 54, =Indag. Math., 13 (1951), 50–60 | DOI | MR | Zbl

[13] N. G. de Bruijn, “On the number of positive integers $\leq x$ and free of prime factors ${>y}$. II”, Nederl. Akad. Wetensch. Proc. Ser. A, 69, =Indag. Math., 28 (1966), 239–247 | DOI | MR | Zbl

[14] L. De Haan, S. I. Resnick, H. Rootzén, C. de Vries, “Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes”, Stochastic Process. Appl., 32:2 (1989), 213–224 | DOI | MR | Zbl

[15] G. Derfel, Yaqin Feng, S. Molchanov, “Probabilistic approach to a cell growth model”, Differential equations, mathematical physics, and applications: Selim Grigorievich Krein centennial, Contemp. Math., 734, Amer. Math. Soc., Providence, RI, 2019, 95–106 | DOI | MR | Zbl

[16] B. Derrida, “Random-energy model: limit of a family of disordered models”, Phys. Rev. Lett., 45:2 (1980), 79–82 | DOI | MR

[17] B. Derrida, “Random-energy model: an exactly solvable model of disordered systems”, Phys. Rev. B (3), 24:5 (1981), 2613–2626 | DOI | MR | Zbl

[18] L. Devroye, Non-uniform random variate generation, Springer-Verlag, New York, 1986, xvi+843 pp. | DOI | MR | Zbl

[19] K. Dickman, “On the frequency of numbers containing prime factors of a certain relative magnitude”, Ark. för Mat. A, 22 (1930), 10, 14 pp. | Zbl

[20] Dickman function, Encyclopedia of mathematics,\par, ed. U. Rehmann https://encyclopediaofmath.org/wiki/Dickman_function

[21] J. Diebolt, D. Guégan, “Tail behaviour of the stationary density of general non-linear autoregressive processes of order 1”, J. Appl. Probab., 30:2 (1993), 315–329 | DOI | MR | Zbl

[22] T. Eisele, “On a third-order phase transition”, Comm. Math. Phys., 90:1 (1983), 125–159 | DOI | MR | Zbl

[23] P. Embrechts, C. Goldie, “Perpetuities and random equations”, Asymptotic statistics (Prague, 1993), Contrib. Statist., Physica, Heidelberg, 1994, 75–86 | DOI | MR

[24] P. Embrechts, C. Klüppelberg, T. Mikosch, Modelling extremal events for insurance and finance, Appl. Math. (N. Y.), 33, Springer-Verlag, Berlin, 1997, xvi+645 pp. | DOI | MR | Zbl

[25] P. Erdös, “On a family of symmetric Bernoulli convolutions”, Amer. J. Math., 61:4 (1939), 974–976 | DOI | MR | Zbl

[26] P. Erdös, “On the smoothness properties of a family Bernoulli convolutions”, Amer. J. Math., 62 (1940), 180–186 | DOI | MR | Zbl

[27] W. Feller, An introduction to probability theory and its applications, v. 1, 2nd ed., John Wiley and Sons, Inc., New York; Chapman and Hall, Ltd., London, 1957, xv+461 pp. | MR | Zbl | Zbl

[28] Shui Feng, The Poisson–Dirichlet distribution and related topics. Models and asymptotic behaviors, Probab. Appl. (N. Y.), Springer, Heidelberg, 2010, xiv+218 pp. | DOI | MR | Zbl

[29] V. Gončarov, “On the field of combinatory analysis”, Amer. Math. Soc. Transl. Ser. 2, 19, Amer. Math. Soc., Providence, RI, 1962, 1–46 | DOI | MR | MR | Zbl | Zbl

[30] M. Grabchak, S. A. Molchanov, “Limit theorems for random exponentials: the bounded support case”, Teoriya veroyatn. i ee primen., 63:4 (2018), 779–794 ; Theory Probab. Appl., 63:4 (2019), 634–647 | DOI | MR | Zbl | DOI

[31] A. Hildebrand, “Integers free of large prime factors and the Riemann hypothesis”, Mathematika, 31:2 (1984), 258–271 | DOI | MR | Zbl

[32] A. Hildebrand, “On the number of positive integers $\leq x$ and free of prime factors ${> y}$”, J. Number Theory, 22:3 (1986), 289–307 | DOI | MR | Zbl

[33] A. Hildebrand, G. Tenenbaum, “On integers free of large prime factors”, Trans. Amer. Math. Soc., 296:1 (1986), 265–290 | DOI | MR | Zbl

[34] A. Hildebrand, G. Tenenbaum, “Integers without large prime factors”, J. Théor. Nombres Bordeaux, 5:2 (1993), 411–484 | DOI | MR | Zbl

[35] M. Kac, Probability and related topics in physical sciences (Boulder, CO, 1957), Lectures in Appl. Math., I, Intersci. Publ., London–New York, 1959, xiii+266 pp. | MR | Zbl

[36] D. E. Knuth, L. Trabb Pardo, “Analysis of a simple factorization algorithm”, Theoret. Comput. Sci., 3:3 (1976/77), 321–348 | DOI | MR | Zbl

[37] V. D. Konakov, S. Menozzi, S. A. Molchanov, “Diffusion processes on solvable groups of upper triangular $2\times 2$ matrices and their approximation”, Dokl. Math., 84:1 (2011), 527–530 | DOI | MR | Zbl

[38] V. Konakov, S. Menozzi, S. Molchanov, “The Brownian motion on $\operatorname{Aff}(\mathbb R)$ and quasi-local theorems”, Probabilistic methods in geometry, topology and spectral theory, Contemp. Math., 739, Amer. Math. Soc., Providence, RI, 2019, 97–125 | DOI | MR | Zbl

[39] H. P. McKean, Jr., Stochastic integrals, Probab. Math. Statist., 5, Academic Press, New York, 1969, xiii+140 pp. | DOI | MR | MR | Zbl | Zbl

[40] S. Molchanov, V. Panov, “Limit theorems for the alloy-type random energy model”, Stochastics, 91:5 (2019), 754–772 | DOI | MR

[41] P. Moree, “Nicolaas Govert de Bruijn, the enchanter of friable integers”, Indag. Math. (N. S.), 24:4 (2013), 774–801 | DOI | MR | Zbl

[42] K. K. Norton, Numbers with small prime factors, and the least $k$th power non-residue, Mem. Amer. Math. Soc., 106, Amer. Math. Soc., Providence, RI, 1971, ii+106 pp. | DOI | MR | Zbl

[43] E. Olivieri, P. Picco, “On the existence of thermodynamics for the random energy model”, Comm. Math. Phys., 96:1 (1984), 125–144 | DOI | MR | Zbl

[44] M. D. Penrose, A. R. Wade, “Random minimal directed spanning trees and Dickman-type distributions”, Adv. in Appl. Probab., 36:3 (2004), 691–714 | DOI | MR | Zbl

[45] M. D. Penrose, A. R. Wade, “On the total length of the random minimal directed spanning tree”, Adv. in Appl. Probab., 38:2 (2006), 336–372 | DOI | MR | Zbl

[46] M. D. Penrose, A. R. Wade, “Limit theorems for random spatial drainage networks”, Adv. in Appl. Probab., 42:3 (2010), 659–688 | DOI | MR | Zbl

[47] Y. Peres, W. Schlag, B. Solomyak, “Sixty years of Bernoulli convolutions”, Fractal geometry and stochastics II (Greifswald/Koserow, 1998), Progr. Probab., 46, Birkhäuser, Basel, 2000, 39–65 | DOI | MR | Zbl

[48] Y. Peres, B. Solomyak, “Absolute continuity of Bernoulli convolutions, a simple proof”, Math. Res. Lett., 3:2 (1996), 231–239 | DOI | MR | Zbl

[49] Yu. V. Prohorov, Yu. A. Rozanov, Probability theory. Basic concepts. Limit theorems. Random processes, Grundlehren Math. Wiss., 157, Springer-Verlag, New York–Heidelberg, 1969, xi+401 pp. | MR | MR | Zbl | Zbl

[50] S. Ramanujan, The lost notebook and other unpublished papers, with an introduction by G. E. Andrews, Springer-Verlag, Berlin; Narosa Publ. House, New Delhi, 1988, xxv+419 pp. | Zbl

[51] V. Ramaswami, “On the number of positive integers less than $x$ and free of prime divisors greater than $x^{c}$”, Bull. Amer. Math. Soc., 55:12 (1949), 1122–1127 | DOI | MR | Zbl

[52] R. A. Rankin, “The difference between consecutive prime numbers”, J. London Math. Soc., s1-13:4 (1938), 242–247 | DOI | MR | Zbl

[53] I. Rodríguez-Iturbe, A. Rinaldo, Fractal river basins. Chance and self-organization, Cambridge Univ. Press, Cambridge, 2001, 570 pp.

[54] L. C. G. Rogers, D. Williams, Diffusions, Markov processes and martingales, v. 2, Cambridge Math. Lib., Itô calculus, Reprint of the 2nd ed., Cambridge Univ. Press, Cambridge, 2000, xiv+480 pp. | DOI | MR | Zbl

[55] W. Schoutens, Lévy processes in finance. Pricing financial derivatives, John Wiley and Sons, Chichester, 2003, 200 pp.

[56] P. Seba, “Markov chain of distances between parked cars”, J. Phys. A, 41:12 (2008), 122003, 5 pp. | DOI | MR | Zbl

[57] B. Solomyak, “On the random series $\sum \pm\lambda^n$ (an Erdös problem)”, Ann. of Math. (2), 142:3 (1995), 611–625 | DOI | MR | Zbl

[58] G. Tenenbaum, Introduction to analytic and probabilistic number theory, Transl. from the French, Grad. Stud. Math., 163, 3rd ed., Amer. Math. Soc., Providence, RI, 2015, xxiv+629 pp. | DOI | MR | Zbl

[59] A. M. Vershik, “The asymptotic distribution of factorizations of natural numbers into prime divisors”, Soviet Math. Dokl., 34 (1987), 57–61 | MR | Zbl

[60] A. M. Vershik, “Does there exist a Lebesgue measure in the infinite-dimensional space?”, Proc. Steklov Inst. Math., 259 (2007), 248–272 | DOI | MR | Zbl

[61] A. M. Vershik, A. A. Shmidt, “Limiting measures arising in the asymptotic theory of symmetric groups. I”, Theory Probab. Appl., 22:1 (1977), 70–85 | DOI | MR | Zbl

[62] A. M. Vershik, A. A. Shmidt, “Limit measures arising in the asymptotic theory of symmetric groups. II”, Theory Probab. Appl., 23:1 (1978), 36–49 | DOI | MR | Zbl