arXiv subjects

Olivier Marchal

Publications and source records attributed to Olivier Marchal.

At least 19 recordsLinked to original sources

arXiv2026

Higher-order methods of radiative transfer in simulations of the epoch of reionisation: Pn versus M1

In current cosmological simulations, the radiative transfer modules generally rely on the M1 approximation, which has some glaring flaws related to its fluid-like behaviour, such as spurious pseudo-sources and loss of directionality when radiation fronts from different directions collide. Pn, another moment-based model used in other fields of physics, may correct these issues. We aim at testing out Pn in an astrophysical setting and compare it to M1, in order to see if it can indeed correct M1's flaws. Also, we want to use Pn's solutions to better pinpoint M1 errors. We implement a Pn radiation transport method and couple it to a photo-thermo-chemistry module to account for the interaction of ionising radiation with the Hydrogen gas, and benchmark it using tests for radiative transfer models comparison in astrophysics as defined in arXiv:astro-ph/0603199. We find that high order P_n (e.g. P9) indeed correct M1's flaws, while faring as well or even better in some aspects in the tests, in particular when directionality is important or colliding radiation fronts occur. By comparing P9 and M1 radiation fields in an idealised and cosmological test case, we highlight a new, thus far unreported artefact of M_1, the 'dark sombrero'. A dark sombrero appears as a spherical photon-deficit shell around the source. The photon density in dark sombreros can be underestimated by a factor up to 2-3. They occur in regions where a source's radiation field connects with that of another source or group of sources. These basic properties (position and amplitude) of the dark sombreros may depend on the sources' relative intensities, positions, spatial resolution, although we have not been able to test this in detail in this study.

astro-ph.CO↗

arXiv2026

Explicit Hamiltonian structure of the Flaschka-Newell Painlevé II hierarchy via symmetry reduction of the Painlevé IV hierarchy

We study the Hamiltonian structure of the Flaschka-Newell Painlevé II hierarchy via symmetry reduction of the associated space of meromorphic connections. Building on the realization of this hierarchy as a reduction of the Painlevé IV hierarchy via a $\mathbb{Z}_2$-symmetry, we construct a set of Darboux coordinates adapted to the involution. After a suitable change of trivialization, the symmetry acts diagonally in these coordinates, allowing the fixed-point locus to be explicitly described as a symplectic submanifold. This enables us to derive the reduced Hamiltonians after symmetry, thereby obtaining explicit expressions for the Hamiltonians and the Lax matrices of the Flaschka-Newell Painlevé II hierarchy. This strategy also illustrates how symmetry-adapted canonical Darboux coordinates enable explicit reductions of isomonodromic systems at the level of their underlying symplectic geometry.

math-ph↗

arXiv2026

A symmetry reduction of the Painlevé IV hierarchy to the Flaschka-Newell Painlevé II hierarchy

We study the isomonodromic deformation problem associated with rank-two meromorphic connections on the Riemann sphere having one regular singularity and one irregular singularity of even order at infinity, corresponding to the even Painlevé IV hierarchy. We show that the symmetry $Ψ(-λ)= σ_1 Ψ(λ) σ_1$ defines an invariant submanifold whose induced isomonodromic dynamics coincides with the Flaschka-Newell Painlevé II hierarchy. Under this identification, the corresponding Lax matrices, Darboux coordinates and Hamiltonian structures can be matched explicitly. In particular, the Hamiltonians of the first members of the Flaschka-Newell hierarchy are recovered from the even Painlevé IV hierarchy. This provides a geometric interpretation of the Flaschka-Newell hierarchy as a symmetry reduction of an isomonodromic deformation problem, complementing its classical description as a similarity reduction of the modified Korteweg-de Vries hierarchy.

math-ph↗

arXiv2026

Geometry of Logarithmic Topological Recursion: Dilaton Equations, Free Energies and Variational Formulas

One of the most important applications of topological recursion concerns spectral curves for which the functions $(x,y)$ defining the spectral curve are allowed to have logarithmic singularities. This occurs for instance for Seiberg-Witten curves and mirror curves computing Gromov--Witten invariants of toric Calabi--Yau threefolds. A recently introduced extension of topological recursion, the so-called logarithmic topological recursion, exhibits the correct behavior under certain limits of those spectral curves. In this article, we derive the dilaton equations in the setting of logarithmic topological recursion, as well as variational formulas, and provide a definition of the free energies in situations where standard topological recursion was known to fail. We present examples in which the new definition of the free energies \textit{directly} (without any computation) reproduces the full perturbative part of the Nekrasov--Shatashvili partition function of 4d $\mathcal{N}=2$ pure supersymmetric gauge theory, as well as the all-genus free energies of mirror curves of strip geometries, including in particular the topological vertex and the resolved conifold.

math-ph↗

arXiv2026

Explicit Hamiltonian representations of meromorphic connections and duality from different perspectives: a case study

In this article, we present an explicit study of $\hbar$-deformed meromorphic connections in $\mathfrak{gl}_3(\mathbb{C})$ with an unramified irregular pole at infinity of order $r_\infty=3$ and its spectral dual corresponding to the $\mathfrak{gl}_2(\mathbb{C})$ Painlevé IV Lax pair. Using the apparent singularities and their dual partners on the spectral curves as Darboux coordinates, we obtain the Hamiltonian evolutions, the reduction of these evolutions to a single non-trivial direction, the Jimbo-Miwa-Ueno tau-functions, the fundamental symplectic two-forms and the associated Hermitian matrix models on both sides. We then prove that the spectral duality connecting both sides extends to all these aspects, providing an explicit illustration of the generalized Harnad duality. We finally propose a conjecture relating the Jimbo-Miwa-Ueno differential as the $\hbar=0$ evaluation of the Hamiltonian differential in these Darboux coordinates that could provide insights on the geometric interpretation of the $\hbar$ formal parameter. As a byproduct we also obtain a rank $3$ Lax pair for the Painlevé IV equation.

math-ph↗

arXiv2026

GW/DT invariants and 5D BPS indices for strips from topological recursion

Topological string theory partition function gives rise to Gromov-Witten invariants, Donaldson-Thomas invariants and 5D BPS indices. Using the remodeling conjecture, which connects Topological Recursion with topological string theory for toric Calabi-Yau threefolds, we study a more direct connection for the subclass of strip geometries. In doing so, new developments in the theory of topological recursion are applied as its extension to Logarithmic Topological Recursion (Log-TR) and the universal $x$-$y$ duality. Through these techniques, our main result in this paper is a direct derivation of all free energies from topological recursion for general strip geometries. In analyzing the expression of free energy, we shed some light on the meaning and the influence of the $x$-$y$ duality in topological string theory and its interconnection to GW and DT invariants as well as the 5D BPS index.

math-ph↗

arXiv2026

Hamiltonian representation of isomonodromic deformations of twisted rational connections: The Painlevé $1$ hierarchy

In this paper, we build the Hamiltonian system and the corresponding Lax pairs associated to a twisted connection in $\mathfrak{gl}_2(\mathbb{C})$ admitting an irregular and ramified pole at infinity of arbitrary degree, hence corresponding to the Painlevé $1$ hierarchy. We provide explicit formulas for these Lax pairs and Hamiltonians in terms of the irregular times and standard $2g$ Darboux coordinates associated to the twisted connection. Furthermore, we obtain a map that reduces the space of irregular times to only $g$ non-trivial isomonodromic deformations. In addition, we perform a symplectic change of Darboux coordinates to obtain a set of symmetric Darboux coordinates in which Hamiltonians and Lax pairs are polynomial. Finally, we apply our general theory to the first cases of the hierarchy: the Airy case $(g=0)$, the Painlevé $1$ case $(g=1)$ and the next two elements of the Painlevé $1$ hierarchy.

math-ph↗

arXiv2025

Optimal sub-Gaussian variance proxy for 3-mass distributions

We investigate the problem of characterizing the optimal variance proxy for sub-Gaussian random variables,whose moment-generating function exhibits bounded growth at infinity. We apply a general characterization method to discrete random variables with equally spaced atoms. We thoroughly study 3-mass distributions, thereby generalizing the well-studied Bernoulli case. We also prove that the discrete uniform distribution over $N$ points is strictly sub-Gaussian. Finally, we provide an open-source Python package that combines analytical and numerical approaches to compute optimal sub-Gaussian variance proxies across a wide range of distributions.

math.ST↗

arXiv2025

Hamiltonian representation of isomonodromic deformations of general rational connections on $\mathfrak{gl}_2(\mathbb{C})$

In this paper, we study and build the Hamiltonian system attached to any $\mathfrak{gl}_2(\mathbb{C})$ meromorphic connection with an arbitrary number of non-ramified poles of arbitrary degrees. In particular, we propose the Lax pairs and Hamiltonian evolutions expressed in terms of irregular times and monodromies associated to the poles as well as $g$ Darboux coordinates defined as the apparent singularities arising in the oper gauge. Moreover, we also provide a reduction of the isomonodromic deformations to a subset of $g$ non-trivial isomonodromic deformations. This reduction is equivalent to a map reducing the set of irregular times to only $g$ non-trivial isomonodromic times. We apply our construction to all cases where the associated spectral curve has genus 1 and recover the standard Painlevé equations. We finally make the connection with the topological recursion and the quantization of classical spectral curve from this perspective.

math-ph↗

arXiv2025

The Painlevé I hierarchy: Correspondence between the isomonodromic approach and the minimal models of the KP hierarchy

Two approaches to the Painlevé I hierarchy are discussed: the isomonodromic construction based on meromorphic connections, and the minimal models construction based on a reduction of the KP hierarchy. An explicit correspondence between both formalisms is built which gives the identification of these setups. In particular, this provides new expressions for the Lax matrices and Hamiltonians.

math-ph↗

arXiv2025

Low Surface Brightness structures from annotated deep CFHT images: effects of the host galaxy's properties and environment

Hierarchical galactic evolution models predict that mergers drive galaxy growth, producing low surface brightness (LSB) tidal features that trace galaxies' late assembly. These faint structures encode information about past mergers and are sensitive to the properties and environment of the host galaxy. We investigated the relationships between LSB features and their hosts in a sample of 475 nearby massive galaxies spanning diverse environments (field, groups, Virgo cluster) using deep optical imaging from the Canada-France-Hawaii Telescope (MATLAS, UNIONS/CFIS, VESTIGE, NGVS). Using Jafar, an online annotation tool, we manually annotated tidal features and extended stellar haloes, including 199 tidal tails and 100 streams. Geometric and photometric measurements were extracted to analyse their dependence on galaxy mass, environment, and internal kinematics. At our surface brightness limit of 29 mag$\,$arcsec$^{-2}$, tidal features and stellar haloes contribute 2% and 10% of total galaxy luminosity, respectively. Tidal features are detected in 36% of galaxies, with none fainter than 27.8 mag$\,$arcsec$^{-2}$. The most massive galaxies are twice as likely to host tidal debris, and for early-type galaxies their halos are twice as luminous as those in lower-mass systems, a trend not observed in late-type galaxies. Although small-scale interactions increase the frequency of tidal features, the large-scale environment does not influence it. An anticorrelation between this frequency and rotational support is found, but may reflect the mass-driven effect. We release our database of annotated features for deep learning applications. Our findings confirm that galaxy mass is the dominant factor influencing tidal feature prevalence, consistent with hierarchical formation models.

astro-ph.GA↗

arXiv2025

Resurgence of Tritronquées Solutions of the Deformed Painlevé I Equation

We prove that the formal $\hbar$-power series solution of the deformed Painlevé I equation is resurgent, which means it is generically Borel summable and its Borel transform admits endless analytic continuation. In particular, we find that the Borel transform defines a global multivalued holomorphic function on a singular algebraic surface isomorphic to the Fermat quintic surface $x^5 + y^5 + z^5 = 0$ modulo an involution. This surface is an algebraic fibration over the complex plane of the differential equation with generic fibre a smooth quintic curve. Each fibre is equipped with a fivefold covering map over another complex plane (the Borel plane) with ten ramification points (the Borel singularities) spread equally over two branch points giving two opposite Stokes rays.

math.DG↗

arXiv2024

Optimal sub-Gaussian variance proxy for truncated Gaussian and exponential random variables

This paper establishes the optimal sub-Gaussian variance proxy for truncated Gaussian and truncated exponential random variables. The proofs rely on first characterizing the optimal variance proxy as the unique solution to a set of two equations and then observing that for these two truncated distributions, one may find explicit solutions to this set of equations. Moreover, we establish the conditions under which the optimal variance proxy coincides with the variance, thereby characterizing the strict sub-Gaussianity of the truncated random variables. Specifically, we demonstrate that truncated Gaussian variables exhibit strict sub-Gaussian behavior if and only if they are symmetric, meaning their truncation is symmetric with respect to the mean. Conversely, truncated exponential variables are shown to never exhibit strict sub-Gaussian properties. These findings contribute to the understanding of these prevalent probability distributions in statistics and machine learning, providing a valuable foundation for improved and optimal modeling and decision-making processes.

math.ST↗

arXiv2024

Isomonodromic and isospectral deformations of meromorphic connections: the $\mathfrak{sl}_2(\mathbb{C})$ case

We consider non-twisted meromorphic connections in $\mathfrak{sl}_2(\mathbb{C})$ and the associated symplectic Hamiltonian structure. In particular, we provide explicit expressions of the Lax pair in the geometric gauge supplementing previous results where explicit formulas have been obtained in the oper gauge. Expressing the geometric Lax matrices requires the introduction of specific Darboux coordinates for which we provide the explicit Hamiltonian evolutions. These expressions allow to build bridges between the isomonodromic deformations and the isospectral ones. More specifically, we propose an explicit change of Darboux coordinates to obtain isospectral coordinates for which Hamiltonians match the spectral invariants. This result solves the issue left opened in \cite{BertolaHarnadHurtubise2022} in the case of $\mathfrak{sl}_2(\mathbb{C})$.

math-ph↗

arXiv2024

Quantization of classical spectral curves via topological recursion

We prove that the topological recursion formalism can be used to quantize any generic classical spectral curve with smooth ramification points and simply ramified away from poles. For this purpose, we build both the associated quantum curve, i.e.~the differential operator quantizing the algebraic equation defining the classical spectral curve considered, and a basis of wave functions, that is to say a basis of solutions of the corresponding differential equation. We further build a Lax pair representing the resulting quantum curve and thus present it as a point in an associated space of meromorphic connections on the Riemann sphere, a first step towards isomonodromic deformations. We finally propose two examples: the derivation of a 2-parameter family of formal trans-series solutions to Painlevé 2 equation and the quantization of a degree three spectral curve with pole only at infinity.

math-ph↗

arXiv2024

Hearing the shape of an arena with spectral swarm robotics

Swarm robotics promises adaptability to unknown situations and robustness against failures. However, it still struggles with global tasks that require understanding the broader context in which the robots operate, such as identifying the shape of the arena in which the robots are embedded. Biological swarms, such as shoals of fish, flocks of birds, and colonies of insects, routinely solve global geometrical problems through the diffusion of local cues. This paradigm can be explicitly described by mathematical models that could be directly computed and exploited by a robotic swarm. Diffusion over a domain is mathematically encapsulated by the Laplacian, a linear operator that measures the local curvature of a function. Crucially the geometry of a domain can generally be reconstructed from the eigenspectrum of its Laplacian. Here we introduce spectral swarm robotics where robots diffuse information to their neighbors to emulate the Laplacian operator - enabling them to "hear" the spectrum of their arena. We reveal a universal scaling that links the optimal number of robots (a global parameter) with their optimal radius of interaction (a local parameter). We validate experimentally spectral swarm robotics under challenging conditions with the one-shot classification of arena shapes using a sparse swarm of Kilobots. Spectral methods can assist with challenging tasks where robots need to build an emergent consensus on their environment, such as adaptation to unknown terrains, division of labor, or quorum sensing. Spectral methods may extend beyond robotics to analyze and coordinate swarms of agents of various natures, such as traffic or crowds, and to better understand the long-range dynamics of natural systems emerging from short-range interactions.

cs.RO↗

arXiv2023

Asymptotic expansion of Toeplitz determinants of an indicator function with discrete rotational symmetry and powers of random unitary matrices

In this short article we propose a full large $N$ asymptotic expansion of the probability that the $m^{\text{th}}$ power of a random unitary matrix of size $N$ has all its eigenvalues in a given arc-interval centered in $1$ when $N$ is large. This corresponds to the asymptotic expansion of a Toeplitz determinant whose symbol is the indicator function of several intervals having a discrete rotational symmetry. This solves and improves a conjecture left opened by the author. It also provides a rare example of the explicit computation of a full asymptotic expansion of a genus $g>0$ classical spectral curve, including the oscillating non-perturbative terms, using the topological recursion.

math.PR↗

arXiv2023

Gaia Data Release 3. Astrometric binary star processing

Context.The Gaia Early Data Release 3 contained the positions, parallaxes and proper motions of 1.5 billion sources, among which some did not fit well the "single star" model. Binarity is one of the causes of this. Aims. Four million of these stars were selected and various models were tested to detect binary stars and to derive their parameters. Methods. A preliminary treatment was used to discard the partially resolved double stars and to correct the transits for perspective acceleration. It was then investigated whether the measurements fit well with an acceleration model with or without jerk. The orbital model was tried when the fit of any acceleration model was beyond our acceptance criteria. A Variability-Induced Mover (VIM) model was also tried when the star was photometrically variable. A final selection has been made in order to keep only solutions that probably correspond to the real nature of the stars. Results. At the end, 338,215 acceleration solutions, about 165,500 orbital solutions and 869 VIM solutions were retained. In addition, formulae for calculating the uncertainties of the Campbell orbital elements from orbital solutions expressed in Thiele-Innes elements are given in an appendix.

astro-ph.SR↗