Description of the thesis topic

Title: Genealogy of extreme particles in branching processes and links with particle physics observables

Branching Brownian motion (BBM) is a system of particles that move continuously and reproduce independently of each other. It is one of the simplest spatial branching processes. The BBM is a model currently much studied in mathematics, due to its links with the famous Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) reaction-diffusion equation (see e.g. [1,2,3]).

Branching processes in the BBM universality class are ubiquitous in the physical and natural sciences. They appear, for example, in spin glass physics, chemistry, evolutionary biology, etc. and, perhaps more surprisingly, also in particle physics. Indeed, the interacting state of quantum particles is generically a random set of elementary particles, which in a certain regime relevant to very high-energy colliders, appear generated by a branching process of the BBM universality class [4]. The total cross sections can be mapped to the statistics of the extreme values of this process [4]. The genealogies of extreme particles [5], the detailed study of which is the subject of this project, correspond to final-state observables [6]. Their qualitative understanding will lead to predictions for electron-nucleus diffractive cross sections, which can be compared with measurements from the future electron-ion collider (EIC) planned in the USA.

The main goals of the proposed PhD thesis will be to describe the genealogical law and relationships of BBM extreme particles, and to establish quantitative correspondences with observables measured in electron-hadron collisions, such as the angular distribution of the diffracted mass in the final state. As the project progresses, we may extend the study to the case of branching-selection processes, which are also relevant in particle physics (see [7,8]). The role of the PhD student will be both to participate in the elaboration of the mathematical proofs, and to develop a physical intuition that will help the formulation of conjectures about the properties of particles generated by a branching process.

The thesis, funded under a project of the CNRS Mission pour les initiatives transverses et interdisciplinaires, will last 3 years, ideally starting on October 1, 2024.

References:

[1] E. Aïdékon, J. Berestycki, É. Brunet, and Z. Shi. Branching Brownian motion seen from its tip. Probab. Theory Relat. Fields, 157(1-2):405–451, 2013.
[2] L. Mytnik, J.-M. Roquejoffre, and Lenya Ryzhik. Fisher-KPP equation with small data and the extremal process of branching Brownian motion. Adv. Math., 396:58, 2022. Id/No 108106.m
[3] P. Maillard and M. Pain. 1-stable fluctuations in branching Brownian motion at critical temperature I: The derivative martingale. Ann. Probab. 47, no. 5, 2953–3002, 2019.
[4] A.-K. Angelopoulou, A. D. Le, and S. Munier. Scattering from an external field in quantum chromodynamics at high energies : from foundations to interdisciplinary connections. arXiv:2311.14796.
[5] B. Derrida and P. Mottishaw. On the genealogy of branching random walks and of directed polymers. EPL,115(4):40005, August 2016.
[6] A. D. Le, A. H. Mueller, and S. Munier. Analytical asymptotics for hard diffraction. Phys. Rev. D, 104:034026, 2021.
[7] É. Brunet, B. Derrida, A. H. Mueller, and S. Munier. Effect of selection on ancestry: An exactly soluble case and its phenomenological generalization. Physical Review E, 76(4), October 2007.
[8] A. Cortines and B. Mallein. A N-branching random walk with random selection. ALEA, Lat. Am. J. Probab. Math. Stat., 14(1):117–137, 14(1):117–137, 2017.

Work Context

The PhD student will be based at École polytechnique's Centre de Physique Théorique (CPHT), where he or she will be supervised by Stéphane Munier. Several long stays are planned at the Institut de Mathématiques de Toulouse (IMT), where they will be supervised by Bastien Mallein (project leader). They will also be able to exchange ideas with several other members of the IMT also members of the MITI project.

The CPHT is a joint unit of the Centre national de la recherche scientifique (CNRS), a major player in fundamental research worldwide and the only French organization active in all scientific fields, and the École polytechnique. École polytechnique is one of France's most selective scientific and technical universities. Located on the campus of the Institut de Polytechnique de Paris (IP Paris) in Palaiseau, 30 minutes from Paris, the CPHT brings together around 100 researchers, including 34 permanent staff, divided into 5 thematic groups: particle physics, string theory, mathematical physics, condensed matter and plasma theory. The PhD student will be integrated into the mathematical physics group.

The IMT is a CNRS joint research unit, and brings together 240 permanent teaching and research staff, engineers, technicians and administrative staff, as well as 120 PhD students and around 30 post-docs on average. It is the largest mathematics laboratory in the south-west of France, based in Toulouse and operating under six supervisory authorities: CNRS, INSA, INUC, Université Toulouse 1 Capitole, Université Toulouse-Jean Jaurès, Université Toulouse 3-Paul Sabatier. The laboratory's research themes cover all mathematical fields, from the most theoretical to the most applied, and are organized around 6 teams corresponding to mathematical sub-disciplines. The PhD student will be attached to the Probability team, whose research themes include the study of functional inequalities, random matrices, branching processes and mathematical physics.

In addition to their research work, doctoral students will take courses offered by the IP Paris doctoral school. They will also have the opportunity to teach if they so wish, and depending on vacancies at the École polytechnique.

Constraints and risks

The thesis will be attached to the Institut Polytechnique de Paris doctoral school (Palaiseau). Several trips to Toulouse are planned as part of the project, including three month-long stays.

Additional Information

The candidate for this thesis will have an excellent master's degree in theoretical physics or mathematics (specializing in probability). They must have a solid grounding in probability and/or advanced statistical mechanics, and an interest in quantum field theory and particle physics. Notions in this field are a definite plus. He or she has a strong interest in rigorous mathematics and its application to particle physics.

Applications should include a detailed CV, a letter of motivation, transcripts of Master's grades, and at least two references (persons likely to be contacted).

The deadline for applications is 28/05/2024.