Nonlinear recombinations and generalized random transpositions
[Recombinaisons non linéaires et transpositions aléatoires généralisées]
Caputo, Pietro1 ; Parisi, Daniel1
1Department of Mathematics and Physics, Roma Tre University, Largo San Murialdo 1, 00146 Roma (Italy)
Annales Henri Lebesgue, Tome 7 (2024), pp. 1245-1299

We study a nonlinear recombination model from population genetics as a combinatorial version of the Kac–Boltzmann equation from kinetic theory. Following Kac’s approach, the nonlinear model is approximated by a mean field linear evolution with a large number of particles. In our setting, the latter takes the form of a generalized random transposition dynamics. Our main results establish a uniform in time propagation of chaos with quantitative bounds, and a tight entropy production estimate for the generalized random transpositions, which holds uniformly in the number of particles. As a byproduct of our analysis we obtain sharp estimates on the speed of convergence to stationarity for the nonlinear equation, both in terms of relative entropy and total variation norm.

Nous étudions un modèle de recombinaison non linéaire issu de la génétique des populations en tant que version combinatoire de l’équation de Kac–Boltzmann de la théorie cinétique. En suivant l’approche de Kac, le modèle non linéaire est approché par une évolution linéaire de champ moyen avec un grand nombre de particules. Dans notre cadre, ce dernier prend la forme d’une dynamique généralisée de transpositions aléatoires. Nos principaux résultats établissent une propagation du chaos uniforme dans le temps, avec des bornes quantitatives, et une estimation de la production d’entropie pour les transpositions aléatoires généralisées, qui est uniforme en le nombre de particules. En conséquence de notre analyse, nous obtenons des estimations précises sur la vitesse de convergence à l’équilibre pour l’équation non linéaire, tant en termes d’entropie relative que de norme de variation totale.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/ahl.219
Classification : 82C40, 82C20, 39B62
Keywords: Nonlinear recombinations, Entropy, Kac program, Permutations, Logarithmic Sobolev inequalities

Caputo, Pietro  1   ; Parisi, Daniel  1

1 Department of Mathematics and Physics, Roma Tre University, Largo San Murialdo 1, 00146 Roma (Italy)
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{AHL_2024__7__1245_0,
     author = {Caputo, Pietro and Parisi, Daniel},
     title = {Nonlinear recombinations and generalized random transpositions},
     journal = {Annales Henri Lebesgue},
     pages = {1245--1299},
     year = {2024},
     publisher = {\'ENS Rennes},
     volume = {7},
     doi = {10.5802/ahl.219},
     zbl = {1547.82024},
     mrnumber = {4799918},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/ahl.219/}
}
TY  - JOUR
AU  - Caputo, Pietro
AU  - Parisi, Daniel
TI  - Nonlinear recombinations and generalized random transpositions
JO  - Annales Henri Lebesgue
PY  - 2024
SP  - 1245
EP  - 1299
VL  - 7
PB  - ÉNS Rennes
UR  - https://www.numdam.org/articles/10.5802/ahl.219/
DO  - 10.5802/ahl.219
LA  - en
ID  - AHL_2024__7__1245_0
ER  - 
%0 Journal Article
%A Caputo, Pietro
%A Parisi, Daniel
%T Nonlinear recombinations and generalized random transpositions
%J Annales Henri Lebesgue
%D 2024
%P 1245-1299
%V 7
%I ÉNS Rennes
%U https://www.numdam.org/articles/10.5802/ahl.219/
%R 10.5802/ahl.219
%G en
%F AHL_2024__7__1245_0
Caputo, Pietro; Parisi, Daniel. Nonlinear recombinations and generalized random transpositions. Annales Henri Lebesgue, Tome 7 (2024), pp. 1245-1299. doi: 10.5802/ahl.219

[BBCB23] Basile, G.; Benedetto, D.; Caglioti, E.; Bertini, L. Large deviations for a binary collision model: energy evaporation, Math. Eng., Volume 5 (2023) no. 1, 1, p. 12 | MR | DOI | Zbl

[BCELM11] Barthe, F.; Cordero-Erausquin, D.; Ledoux, M.; Maurey, B. Correlation and Brascamp–Lieb inequalities for Markov semigroups, Int. Math. Res. Not., Volume 2011 (2011) no. 10, pp. 2177-2216 | MR | DOI | Zbl

[BGLR18] Bonetto, F.; Geisinger, A.; Loss, M.; Ried, T. Entropy decay for the Kac evolution, Commun. Math. Phys., Volume 363 (2018) no. 3, pp. 847-875 | MR | DOI | Zbl

[BT06] Bobkov, S. G.; Tetali, P. Modified logarithmic Sobolev inequalities in discrete settings, J. Theor. Probab., Volume 19 (2006) no. 2, pp. 289-336 | MR | DOI | Zbl

[CAM02] Caputo, P.; Martinelli, F. Asymmetric diffusion and the energy gap above the 111 ground state of the quantum XXZ model, Commun. Math. Phys., Volume 226 (2002) no. 2, pp. 323-375 | MR | DOI | Zbl

[CAS18] Caputo, P.; Sinclair, A. Entropy production in nonlinear recombination models, Bernoulli, Volume 24 (2018) no. 4B, pp. 3246-3282 | MR | DOI | Zbl

[CCG00] Carlen, E. A.; Carvalho, M. C.; Gabetta, E. Central limit theorem for Maxwellian molecules and truncation of the Wild expansion, Commun. Pure Appl. Math., Volume 53 (2000) no. 3, pp. 370-397 | MR | Zbl | DOI

[CCLR + 10] Carlen, E. A.; Carvalho, M. C.; Le Roux, J.; Loss, M.; Villani, C. Entropy and chaos in the Kac model, Kinet. Relat. Models, Volume 3 (2010) no. 1, pp. 85-122 | MR | DOI | Zbl

[CD22] Chaintron, L.-P.; Diez, A. Propagation of chaos: a review of models, methods and applications. I: Models and methods, Kinet. Relat. Models, Volume 15 (2022) no. 6, pp. 895-1015 | MR | DOI | Zbl

[CGM08] Cattiaux, P.; Guillin, A.; Malrieu, F. Probabilistic approach for granular media equations in the non-uniformly convex case, Probab. Theory Relat. Fields, Volume 140 (2008) no. 1-2, pp. 19-40 | MR | DOI | Zbl

[Chu00] Chung, K. L. A course in probability theory, Academic Press Inc., 2000 | MR | Zbl

[CHV20] Csóka, E.; Harangi, V.; Virág, B. Entropy and expansion, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 56 (2020) no. 4, pp. 2428-2444 | MR | DOI | Zbl

[DEGZ20] Durmus, A.; Eberle, A.; Guillin, A.; Zimmer, R. An elementary approach to uniform in time propagation of chaos, Proc. Am. Math. Soc., Volume 148 (2020) no. 12, pp. 5387-5398 | MR | DOI | Zbl

[DGR09] Dolera, E.; Gabetta, E.; Regazzini, E. Reaching the best possible rate of convergence to equilibrium for solutions of Kac’s equation via central limit theorem, Ann. Appl. Probab., Volume 19 (2009) no. 1, pp. 186-209 | MR | DOI | Zbl

[DMG01] Del Moral, P.; Guionnet, A. On the stability of interacting processes with applications to filtering and genetic algorithms, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 37 (2001) no. 2, pp. 155-194 | MR | DOI | Zbl | Numdam

[DMM01] Del Moral, P.; Miclo, L. Genealogies and increasing propagation of chaos for Feynman–Kac and genetic models, Ann. Appl. Probab., Volume 11 (2001) no. 4, pp. 1166-1198 | MR | Zbl

[DS81] Diaconis, P.; Shahshahani, M. Generating a random permutation with random transpositions, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 57 (1981), pp. 159-179 | MR | DOI | Zbl

[DSC96] Diaconis, P.; Saloff-Coste, L. Logarithmic Sobolev inequalities for finite Markov chains, Ann. Appl. Probab., Volume 6 (1996) no. 3, pp. 695-750 | MR | DOI | Zbl

[EFS20] Erbar, M.; Fathi, M.; Schlichting, A. Entropic curvature and convergence to equilibrium for mean-field dynamics on discrete spaces, ALEA, Lat. Am. J. Probab. Math. Stat., Volume 17 (2020) no. 1, pp. 445-471 | MR | Zbl | DOI

[Ein11] Einav, A. On Villani’s conjecture concerning entropy production for the Kac master equation, Kinet. Relat. Models, Volume 4 (2011) no. 2, pp. 479-497 | MR | DOI | Zbl

[Gei44] Geiringer, H. On the probability theory of linkage in Mendelian heredity, Ann. Math. Stat., Volume 15 (1944), pp. 25-57 | MR | DOI | Zbl

[Goe04] Goel, S. Modified logarithmic Sobolev inequalities for some models of random walk, Stochastic Processes Appl., Volume 114 (2004) no. 1, pp. 51-79 | MR | DOI | Zbl

[GQ03] Gao, F.; Quastel, J. Exponential decay of entropy in the random transposition and Bernoulli–Laplace models, Ann. Appl. Probab., Volume 13 (2003) no. 4, pp. 1591-1600 | MR | DOI | Zbl

[Grü71] Grünbaum, F. A. Propagation of chaos for the Boltzmann equation, Arch. Ration. Mech. Anal., Volume 42 (1971), pp. 323-345 | MR | DOI | Zbl

[HM14] Hauray, M.; Mischler, S. On Kac’s chaos and related problems, J. Funct. Anal., Volume 266 (2014) no. 10, pp. 6055-6157 | MR | DOI | Zbl

[Kac56] Kac, M. Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, University of California Press (1956), pp. 171-197 | Zbl

[Lac23] Lacker, D. Hierarchies, entropy, and quantitative propagation of chaos for mean field diffusions, Probab. Math. Phys., Volume 4 (2023) no. 2, pp. 377-432 | MR | DOI | Zbl

[Lyu92] Lyubich, Y. I. Mathematical structures in population genetics, Biomathematics (Berlin), 22, Springer, 1992 (translated from the Russian by D. Vulis and A. Karpov. Edited by E. Akin) | MR | Zbl

[Mar17] Martínez, S. A probabilistic analysis of a discrete-time evolution in recombination, Adv. Appl. Math., Volume 91 (2017), pp. 115-136 | DOI | Zbl

[McK66] McKean, H. P. jun. Speed of approach to equilibrium for Kac’s caricature of a Maxwellian gas, Arch. Ration. Mech. Anal., Volume 21 (1966), pp. 343-367 | MR | DOI | Zbl

[McK67] McKean, H. P. jun. An exponential formula for solving Boltzmann’s equation for a Maxwellian gas, Comb. Theory, Volume 2 (1967), pp. 358-382 | MR | DOI | Zbl

[MM13] Mischler, S.; Mouhot, C. Kac’s program in kinetic theory, Invent. Math., Volume 193 (2013) no. 1, pp. 1-147 | MR | DOI | Zbl

[MMW15] Mischler, S.; Mouhot, C.; Wennberg, B. A new approach to quantitative propagation of chaos for drift, diffusion and jump processes, Probab. Theory Relat. Fields, Volume 161 (2015) no. 1-2, pp. 1-59 | MR | DOI | Zbl

[Nag13] Nagylaki, T. Introduction to theoretical population genetics, Biomathematics (Berlin), 21, Springer, 2013 | MR | Zbl

[Pet22] Petrov, V. V. Sums of independent random variables, Walter de Gruyter, 2022 | DOI

[PI88] Platkowski, T.; Illner, R. Discrete velocity models of the Boltzmann equation: a survey on the mathematical aspects of the theory, SIAM Rev., Volume 30 (1988) no. 2, pp. 213-255 | MR | DOI | Zbl

[Rez96] Rezakhanlou, F. Propagation of chaos for particle systems associated with discrete Boltzmann equatio, Stochastic Processes Appl., Volume 64 (1996) no. 1, pp. 55-72 | MR | DOI | Zbl

[RRS98] Rabani, Y.; Rabinovich, Y.; Sinclair, A. A computational view of population genetics, Random Struct. Algorithms, Volume 12 (1998) no. 4, pp. 313-334 | MR | Zbl | DOI

[RSW92] Rabinovich, Y.; Sinclair, A.; Wigderson, A. Quadratic dynamical systems, 33rd annual symposium on Foundations of computer science (FOCS)Proceedings, IEEE (1992), pp. 304-313 | Zbl | DOI

[SBB16] Salamat, M.; Baake, M.; Baake, E. The general recombination equation in continuous time and its solution, Discrete Contin. Dyn. Syst., Volume 36 (2016) no. 1, pp. 63-95 | MR | DOI | Zbl

[Szn89] Sznitman, A.-S. Topics in propagation of chaos, Calcul des probabilités, Ec. d’Été, Saint-Flour/Fr (Lecture Notes in Mathematics), Volume 1464, Springer (1989), pp. 165-251 | Zbl

[Tan78] Tanaka, H. Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 46 (1978), pp. 67-105 | MR | DOI | Zbl

[Vil03] Villani, C. Cercignani’s conjecture is sometimes true and always almost true, Commun. Math. Phys., Volume 234 (2003) no. 3, pp. 455-490 | MR | DOI | Zbl

[Wil51] Wild, E. On Boltzmann’s equation in the kinetic theory of gases, Proc. Camb. Philos. Soc., Volume 47 (1951) no. 3, pp. 602-609 | MR | Zbl | DOI

Cité par Sources :