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Damien Simon

Publications and source records attributed to Damien Simon.

16 recordsLinked to original sources
arXiv2026

Data-Driven Optimisation of Superconducting Magnets at CEA Paris-Saclay

Superconducting magnets for particle accelerators are particularly challenging to design because they involve a large number of coupled physical phenomena and the management of complex datasets. Artificial Intelligence (AI), including machine learning and advanced optimisation techniques, offers promising approaches to address these challenges and accelerate the design process. This paper presents a new AI-based optimisation and data management platform, and highlights several ongoing applications of AI methods carried out at CEA Paris-Saclay, including multiphysics optimisation using active learning, topology optimisation, holistic modelling of an Electron Cyclotron Resonance (ERC) ion source, and anomaly detection in quench events.

physics.acc-ph↗
arXiv2025

Representation theory of the principal equivariant $\mathcal{W}$-algebra and Langlands duality

We study the structure and the representation theory of a certain class of vertex algebras. Our study was partly motivated by the quantum geometric Langlands program and we explain what some of our results mean in this framework. We begin our investigation with the vertex algebra of chiral differential operators on a reductive group $\mathcal{D}_{G}^{\kappa}$ for generic levels. In particular, we prove that its vertex algebraic structure is essentially unique. We also study its representation theory and show that the geometric Satake equivalence degenerates. The latter leads us to formulate a vertex-algebraic version of the fundamental local equivalence of Gaitsgory and Lurie. In turn, this brings us to study the representation theory of the principal equivariant affine $\mathcal{W}$-algebra $\mathcal{W}_{G}^{\kappa}$, defined by Arakawa as the principal quantum Hamiltonian reduction of $\mathcal{D}_{G}^{\kappa}$. We construct a family of simple modules for $\mathcal{W}_{G}^{\kappa}$ whose combinatorics matches that of the representation theory of the Langlands dual group. Finally, we establish the fundamental local equivalence when the group is an algebraic torus or is simple adjoint of classical simply laced type.

math.RT↗
arXiv2024

Mixed radix numeration bases: Horner's rule, Yang-Baxter equation and Furstenberg's conjecture

Mixed radix bases in numeration is a very old notion but it is rarely studied on its own or in relation with concrete problems related to number theory. Starting from the natural question of the conversion of a basis to another for integers as well as polynomials, we use mixed radix bases to introduce two-dimensional arrays with suitable filling rules. These arrays provide algorithms of conversion which uses only a finite number of euclidean division to convert from one basis to another; it is interesting to note that these algorithms are generalizations of the well-known Horner's rule of quick evaluation of polynomials. The two-dimensional arrays with local transformations are reminiscent from statistical mechanics models: we show that changes between three numeration basis are related to the set-theoretical Yang-Baxter equation and this is, up to our knowledge, the first time that such a structure is described in number theory. As an illustration, we reinterpret well-known results around Furstenberg's conjecture in terms of Yang-Baxter transformations between mixed radix bases, hence opening the way to alternative approaches.

math-ph↗
arXiv2023

Operadic structure of boundary conditions for two-dimensional Markov Gaussian random fields on the lattice

The theory of Markov processes on the square lattice has been given recently by the second author a new algebraic description in terms of operads. In particular, this new approach allows for a nice description of invariant boundary conditions and infinite-volume Gibbs measures. This theory comes with new algebraic objects which have not been constructed on any non trivial model yet. In this article, the main objective is to exhibit and understand these structures in the particular case of Gaussian Markov fields on the two-dimensional square lattice. This article, in the Gaussian framework, is the first time where all the operadic constructions -- products and eigen-elements up to morphisms -- introduced by the second author are defined rigorously. We also relate these constructions to more classical approach such as the transfer matrix of statistical mechanics and the Fourier transform. The description of half-strips and corners is new and requires the introduction of new operations such as folding. From the probabilistic point of view, we also show that the operadic products on the boundaries are not easily defined and most operations are lifted to the level of parameter spaces, here quadratic forms through Schur complements.

math.PR↗
arXiv2023

Operads and the Markov Property on the square lattice

Markov processes on the lattices with arbitrary dimension are omnipresent in statistical mechanics; however their algebraic description is complete only in dimension 1, for which linear algebra provides many tools complementary to the probabilistic approach: as an example, invariant measures are eigenvectors of the generator. In larger dimension, such algebraic tools are absent due to the more involved structure of boundaries. The present work fills this gap by providing a new and complete algebraic description of these models without any other assumption than the Markov property. In order to handle higher dimensions and higher products, the language of operads is used in order to focus on associativities and the geometric interpretation of the algebraic products. This formalism leads to a new parametrization of boundary conditions of Markov processes, with concrete computations. This parametrization is inspired matrix product states in the physics literature. Among others, a probabilistic application elaborated in this paper is the construction of translation-invariant infinite-volume Gibbs measures on the whole lattice by using Kolmogorov's extension; this provides a new alternative tool to the traditional analytical approaches of large size limits. Various models are considered as illustrations.

math.PR↗
arXiv2020

Area anomaly in the rough path Brownian scaling limit of hidden Markov walks

We study the convergence in rough path topology of a certain class of discrete processes, the hidden Markov walks, to a Brownian motion with an area anomaly. This area anomaly, which is a new object, keeps track of the time-correlation of the discrete models and brings into light the question of embeddings of discrete processes into continuous time. We also identify an underlying combinatorial structure in the hidden Markov walks, which turns out to be a generalization of the occupation time from the classical ergodic theorem in the spirit of rough paths.

math.PR↗
arXiv2019

Random knots in three-dimensional three-colour percolation: numerical results and conjectures

Three-dimensional three-colour percolation on a lattice made of tetrahedra is a direct generalization of two-dimensional two-colour percolation on the triangular lattice. The interfaces between one-colour clusters are made of bicolour surfaces and tricolour non-intersecting and non-self-intersecting curves. Because of the three-dimensional space, these curves describe knots and links. The present paper presents a construction of such random knots using particular boundary conditions and a numerical study of some invariants of the knots. The results are sources of precise conjectures about the limit law of the Alexander polynomial of the random knots.

math-ph↗
arXiv2017

Lévy area with a drift as a renormalization limit of Markov chains on periodic graphs

A careful look at rough path topology applied to Brownian motion reveals new possible properties of the well-known Lévy area, in particular the presence of an intrinsic drift of this area. Using renormalization limit of Markov chains on periodic graphs, we present a construction of such a non-trivial drift and give an explicit formula for it. Several examples for which explicit computations are made are included.

math.PR↗
arXiv2011

Bethe Ansatz for the Weakly Asymmetric Simple Exclusion Process and phase transition in the current distribution

The probability distribution of the current in the asymmetric simple exclusion process is expected to undergo a phase transition in the regime of weak asymmetry of the jumping rates. This transition was first predicted by Bodineau and Derrida using a linear stability analysis of the hydrodynamical limit of the process and further arguments have been given by Mallick and Prolhac. However it has been impossible so far to study what happens after the transition. The present paper presents an analysis of the large deviation function of the current on both sides of the transition from a Bethe ansatz approach of the weak asymmetry regime of the exclusion process.

cond-mat.stat-mech↗
arXiv2010

Eigenvectors of open XXZ and ASEP models for a class of non-diagonal boundary conditions

We present a generalization of the coordinate Bethe ansatz that allows us to solve integrable open XXZ and ASEP models with non-diagonal boundary matrices, provided their parameters obey some relations. These relations extend the ones already known in the literature in the context of algebraic or functional Bethe ansatz. The eigenvectors are represented as sums over cosets of the $BC_n$ Weyl group.

cond-mat.stat-mech↗
arXiv2010

Asymmetric simple exclusion process on a ring conditioned on enhanced flux

We show that in the asymmetric simple exclusion process (ASEP) on a ring, conditioned on carrying a large flux, the particle experience an effective long-range potential which in the limit of very large flux takes the simple form $U= -2\sum_{i\neq j}\log|\sinπ(n_{i}/L-n_{j}/L)|$, where $n_{1}% n_{2},\ldots n_{N}$ are the particle positions, similar to the effective potential between the eigenvalues of the circular unitary ensemble in random matrices. Effective hopping rates and various quasistationary probabilities under such a conditioning are found analytically using the Bethe ansatz and determinantal free fermion techniques. Our asymptotic results extend to the limit of large current and large activity for a family of reaction-diffusion processes with on-site exclusion between particles. We point out an intriguing generic relation between classical stationary probability distributions for conditioned dynamics and quantum ground state wave functions, in particular, in the case of exclusion processes, for free fermions.

cond-mat.stat-mech↗
arXiv2009

The speed of evolution in large asexual populations

We consider an asexual biological population of constant size $N$ evolving in discrete time under the influence of selection and mutation. Beneficial mutations appear at rate $U$ and their selective effects $s$ are drawn from a distribution $g(s)$. After introducing the required models and concepts of mathematical population genetics, we review different approaches to computing the speed of logarithmic fitness increase as a function of $N$, $U$ and $g(s)$. We present an exact solution of the infinite population size limit and provide an estimate of the population size beyond which it is valid. We then discuss approximate approaches to the finite population problem, distinguishing between the case of a single selection coefficient, $g(s) = δ(s - s_b)$, and a continuous distribution of selection coefficients. Analytic estimates for the speed are compared to numerical simulations up to population sizes of order $10^{300}$.

q-bio.PE↗
arXiv2009

Construction of a Coordinate Bethe Ansatz for the asymmetric simple exclusion process with open boundaries

The asymmetric simple exclusion process with open boundaries, which is a very simple model of out-of-equilibrium statistical physics, is known to be integrable. In particular, its spectrum can be described in terms of Bethe roots. The large deviation function of the current can be obtained as well by diagonalizing a modified transition matrix, that is still integrable: the spectrum of this new matrix can be also described in terms of Bethe roots for special values of the parameters. However, due to the algebraic framework used to write the Bethe equations in the previous works, the nature of the excitations and the full structure of the eigenvectors were still unknown. This paper explains why the eigenvectors of the modified transition matrix are physically relevant, gives an explicit expression for the eigenvectors and applies it to the study of atypical currents. It also shows how the coordinate Bethe Ansatz developped for the excitations leads to a simple derivation of the Bethe equations and of the validity conditions of this Ansatz. All the results obtained by de Gier and Essler are recovered and the approach gives a physical interpretation of the exceptional points The overlap of this approach with other tools such as the matrix Ansatz is also discussed. The method that is presented here may be not specific to the asymmetric exclusion process and may be applied to other models with open boundaries to find similar exceptional points.

cond-mat.stat-mech↗
arXiv2008

Quasi-stationary regime of a branching random walk in presence of an absorbing wall

A branching random walk in presence of an absorbing wall moving at a constant velocity $v$ undergoes a phase transition as the velocity $v$ of the wall varies. Below the critical velocity $v_c$, the population has a non-zero survival probability and when the population survives its size grows exponentially. We investigate the histories of the population conditioned on having a single survivor at some final time $T$. We study the quasi-stationary regime for $v<v_c$ when $T$ is large. To do so, one can construct a modified stochastic process which is equivalent to the original process conditioned on having a single survivor at final time $T$. We then use this construction to show that the properties of the quasi-stationary regime are universal when $v\to v_c$. We also solve exactly a simple version of the problem, the exponential model, for which the study of the quasi-stationary regime can be reduced to the analysis of a single one-dimensional map.

cond-mat.stat-mech↗
arXiv2006

Evolution of the most recent common ancestor of a population with no selection

We consider the evolution of a population of fixed size with no selection. The number of generations $G$ to reach the first common ancestor evolves in time. This evolution can be described by a simple Markov process which allows one to calculate several characteristics of the time dependence of $G$. We also study how $G$ is correlated to the genetic diversity.

cond-mat.stat-mech↗