The Dickman subordinator, renewal theorems, and disordered systems.

*(English)*Zbl 07107385Summary: We consider the so-called Dickman subordinator, whose Lévy measure has density \(\frac{1} {x}\) restricted to the interval \((0,1)\). The marginal density of this process, known as the Dickman function, appears in many areas of mathematics, from number theory to combinatorics. In this paper, we study renewal processes in the domain of attraction of the Dickman subordinator, for which we prove local renewal theorems. We then present applications to marginally relevant disordered systems, such as pinning and directed polymer models, and prove sharp second moment estimates on their partition functions.

##### MSC:

60K05 | Renewal theory |

82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |

60G51 | Processes with independent increments; Lévy processes |

##### Keywords:

Dickman subordinator; Dickman function; renewal process; levy process; renewal theorem; stable process; disordered system; pinning model; directed polymer model##### References:

[1] | [AKQ14a] T. Alberts, K. Khanin, and J. Quastel. The intermediate disorder regime for directed polymers in dimension \(1+1\). Ann. Probab. 42 (2014), 1212-1256. |

[2] | [AKQ14b] T. Alberts, K. Khanin, and J. Quastel. The Continuum Directed Random Polymer. J. Stat. Phys. 154 (2014), 305-326. |

[3] | [AB16] K. Alexander, Q. Berger. Local limit theorem and renewal theory with no moments. Electron. J. Probab. Vol. 21 (2016), no. 66, 1-18. |

[4] | [ABT03] R. Arratia, A.D. Barbour, S. Tavaré. Logarithmic Combinatorial Structures: a Probabilistic Approach. EMS Monographs in Mathematics (2003). |

[5] | [B19+] Q. Berger. Notes on random walks in the cauchy domain of attraction. Probab. Theory Relat. Fields (to appear). Preprint (2017), arXiv:1706.07924. |

[6] | [BL17] Q. Berger, H. Lacoin. The high-temperature behavior for the directed polymer in dimension \(1+2\). Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017), 430-450. |

[7] | [BL18] Q. Berger, H. Lacoin. Pinning on a defect line: characterization of marginal disorder relevance and sharp asymptotics for the critical point shift. J. Inst. Math. Jussieu 17 (2018), 305-346. |

[8] | [BC98] L. Bertini and N. Cancrini. The two-dimensional stochastic heat equation: renormalizing a multiplicative noise. J. Phys. A: Math. Gen. 31 (1998), 615. |

[9] | [Ber96] J. Bertoin, Lévy Processes, Cambridge University Press (1996). |

[10] | [BGT89] N.H. Bingham, C.H. Goldie and J.L. Teugels. Regular Variation. Cambridge University Press, 1989. |

[11] | [BKKK14] J. Burridge, A. Kuznetsov, M. Kwaśnicki, A.E. Kyprianou. New families of subordinators with explicit transition probability semigroup. Stochastic Process. Appl. 124 (2014), 3480-3495. |

[12] | [CD19] F. Caravenna, R. Doney. Local large deviations and the strong renewal theorem. Electron. J. Probab. 24 (2019), paper no. 72, 1-48. |

[13] | [CSZ17a] F. Caravenna, R. Sun, N. Zygouras. Polynomial chaos and scaling limits of disordered systems. J. Eur. Math. Soc. 19 (2017), 1-65. |

[14] | [CSZ17b] F. Caravenna, R. Sun, N. Zygouras. Universality in marginally relevant disordered systems. Ann. Appl. Probab. 27 (2017), 3050-3112. |

[15] | [CSZ18] F. Caravenna, R. Sun, N. Zygouras. On the moments of the \((2+1)\)-dimensional directed polymer and stochastic heat equation in the critical window. Preprint (2016), arXiv:1808.03586. |

[16] | [D30] K. Dickman. On the frequency of numbers containing prime factors of a certain relative magnitude. Ark. Math. Astr. Fys. 22 (1930), 1-14. |

[17] | [D97] R.A. Doney. One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Relat. Fields 107 (1997), 451-465. |

[18] | [E70] K.B. Erickson, Strong renewal theorems with infinite mean, Transactions of the AMS 151 (1970), 263-291. |

[19] | [GL62] A. Garsia and J. Lamperti. A discrete renewal theorem with infinite mean. Comm. Math. Helv. 37 (1962), 221-234. |

[20] | [G07] G. Giacomin. Random Polymer Models. Imperial College Press, London, 2007. |

[21] | [G10] G. Giacomin. Disorder and critical phenomena through basic probability models. École d’Été de Probabilités de Saint-Flour XL - 2010. Springer Lecture Notes in Mathematics 2025. |

[22] | [GLT10] G. Giacomin, H. Lacoin, F.L. Toninelli. Marginal relevance of disorder for pinning models. Comm. Pure Appl. Math. 63 (2010) 233-265. |

[23] | [GR07] I.S. Gradshtein and I.M. Ryzhik. Table of Integrals, Series, and Products. Academic Press, Seventh edition (2007). |

[24] | [HT01] H. Hwang, T.-H. Tsai. Quickselect and the Dickman function. Combinatorics, Probability and Computing 11 (2002), 353-371. |

[25] | [K77] J.F.C. Kingman. The population structure associated with the Ewens sampling formula. Theoretical Population Biology 11 (1977), 274-283. |

[26] | [N79] S.V. Nagaev. Large deviations of sums of independent random variables. Ann. Probab. 7 (1979), 745-789. |

[27] | [N12] S.V. Nagaev. The renewal theorem in the absence of power moments. Theory of Probability & Its Applications 56 (2012), 166-175. |

[28] | [NW08] S.V. Nagaev, V.I. Vachtel. On sums of independent random variables without power moments. Siberian Math. Journal 49 (2008), 1091-1100. |

[29] | [RW02] L. Rüschendorf, J.H.C. Woerner. Expansion of transition distributions of Lévy processes in small time. Bernoulli 8 (2002), 81-96. |

[30] | [S99] K.-I. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press (1999). |

[31] | [S09] J. Sohier. Finite size scaling for homogeneous pinning models. ALEA 6 (2009), 163-177. |

[32] | [T95] G. Tenenbaum. Introduction to Analitic and Probabilistic Number Theory. Cambridge University Press (1995). |

[33] | [U11] K. Uchiyama. The first hitting time of a single point for random walks. Electron. J. Probab. 16 (2011), paper n. 71, 1960-2000. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.