arXiv subjects

Jeremie Unterberger

Publications and source records attributed to Jeremie Unterberger.

At least 19 recordsLinked to original sources

arXiv2026

Network analysis reveals phase transitions in agro-food nitrogen systems with contrasting feeding capacity and land requirement

Agro-food transitions are commonly assessed using optimization models or scenario approaches that return a single feasible configuration. These approaches provide estimates of system performance but limited information on feasible nitrogen cycling configurations. In this study, we develop a network-based modelling framework to characterize alternative steady-state configurations of agro-food nitrogen systems and identify phase transitions between these configurations. We formulate a continuous-time compartmental network model in which production, allocation, recycling, and losses are represented as mass-conserving flows. The system is analyzed under steady-state conditions. A coarse-graining procedure identifies dominant recycling cycles and derives aggregate indicators describing system configuration. The model is parameterized using French reference data and applied to case studies covering fertilization, livestock, and dietary patterns. The analysis identifies distinct steady-state configurations of the nitrogen network, separated by phase transitions in dominant recycling structures. These transitions correspond to changes in system performance indicators. For the French reference system, cropland-based feeding capacity is about 8.7 people per hectare of cropland (depending on constraints), while strict dietary autonomy requires about 19.7 million hectares of agricultural land. Reducing synthetic fertilization without changes in diet or nitrogen sourcing leads to higher land requirements or greater dependence on external inputs. The framework provides a method for mapping feasible nitrogen system configurations and analyzing phase transitions in agro-food nitrogen networks under alternative parameters and constraints.

math.OC↗

arXiv2025

General multi-scale estimates for Lyapunov data of Perron-Frobenius matrices. The case of diluted autocatalytic chemical reaction networks

Autocatalytic chemical reaction networks are dynamical systems whose linearization around zero, dX/dt = AX, is represented by a Perron-Frobenius matrix A with positive Lyapunov exponent; this exponent gives the growth rate of the species concentration vector X in the diluted regime, i.e. in a vicinity of zero. We introduce here a new, general recursive procedure providing precise quantitative information about Lyapunov data, namely, the Lyapunov eigenvalue, and left and right eigenvectors. Our estimates are based on a multi-scale algorithm inspired from Wilson's renormalization group method in quantum field theory, and Markov chain arguments introduced in (Nghe & Unterberger). They are compatible with the very scarce knowledge of kinetic rates (coefficients of A) generally available in chemistry, and take on the form of simple rational functions of the latter.

math.PR↗

arXiv2025

Autocatalytic cores in the diluted regime:classification and properties

Autocatalysis underlies the ability of chemical and biochemical systems to replicate. Autocatalysis was recently defined stoichiometrically for reaction networks; five types of minimal autocatalytic networks, termed autocatalytic cores were identified. A necessary and sufficient stoichiometric criterion was later established for dynamical autocatalysis in diluted regimes, ensuring a positive growth rate of autocatalytic species starting from infinitesimal concentrations, given that degradation rates are sufficiently low. Here, we show that minimal autocatalytic networks in the dynamical sense, in the diluted regime, follow the same classification as autocatalytic cores in the stoichiometric sense. We further prove the uniqueness of the stationary regimes of autocatalytic cores, with and without degradation, for all types, except types II with three catalytic loops or more. These results indicate that the stationary point is likely to be robust under perturbation at low concentrations. More complex behaviours are likely to arise by additional non-linear couplings between cores.

q-bio.MN↗

arXiv2025

An introduction to random rule-based chemical networks

Large chemical networks appear in various branches of chemistry and biology, in particular, cellular metabolism and prebiotic chemistry. Detailed simulations of such networks are difficult, or even impossible for lack of kinetic data. Various strategies have been developed to produce synthetic random networks mimicking the large scale organizational properties of experimental chemical networks. These random networks are however mathematical artefacts, which fail to reflect the general reactivity structure of chemistry. We present here a new class of random models of prebiotic (uncatalyzed) chemistry, based on context-independent rules, which is coherent with the general compositional logic of metabolism. The general organization of the random networks fits within the small world paradigm. We get a phase diagram of the models through an approximate mapping to a solvable tree growth model. Our predictions go beyond a purely abstract connectivity analysis of the reaction graph by studying the diversity of chemical mechanisms, and singling out evolutionary patterns, such as autocatalysis and multistationarity, paving the road to possible open-ended evolution.

math.PR↗

arXiv2025

Optimal multi-time-scale estimates for diluted autocatalytic chemical networks. (1) Introduction and $\sigma^*$-dominant case

Autocatalytic chemical networks are dynamical systems whose linearization around zero has a positive Lyapunov exponent; this exponent gives the growth rate of the system in the diluted regime, i.e. for near-zero concentrations. The generator of the dynamics in the kinetic limit is then a Perron-Frobenius matrix, suggesting the use of Markov chain techniques to get long-time asymptotics. This series of works introduces a new, general procedure providing precise quantitative information about such asymptotics, based on estimates for the Lyapunov eigenvalue and eigenvector. The algorithm, inspired from Wilson's renormalization group method in quantum field theory, is based on a downward recursion on kinetic scales, starting from the fastest, and terminating with the slowest rates. Estimates take on the form of simple rational functions of kinetic rates. They are accurate under a separation of scales hypothesis, loosely stating that kinetic rates span many orders of magnitude. We provide here a brief general motivation and introduction to the method, present some simple examples, and derive a number of preliminary results, in particular the estimation of Lyapunov data for a subclass of so-called $\sigma^*$-dominant graphs.

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arXiv2022

Stoechiometric and dynamical autocatalysis for diluted chemical reaction networks

Autocatalysis underlies the ability of chemical and biochemical systems to replicate. Recently, Blokhuis et al. gave a stoechiometric definition of autocatalysis for reaction networks, stating the existence of a combination of reactions such that the balance for all autocatalytic species is strictly positive, and investigated minimal autocatalytic networks, called {\em autocatalytic cores}. By contrast, spontaneous autocatalysis -- namely, exponential amplification of all species internal to a reaction network, starting from a diluted regime, i.e. low concentrations -- is a dynamical property. We introduce here a topological condition (Top) for autocatalysis, namely: restricting the reaction network description to highly diluted species, we assume existence of a strongly connected component possessing at least one reaction with multiple products (including multiple copies of a single species). We find this condition to be necessary and sufficient for stoechiometric autocatalysis. When degradation reactions have small enough rates, the topological condition further ensures dynamical autocatalysis, characterized by a strictly positive Lyapunov exponent giving the instantaneous exponential growth rate of the system. The proof is generally based on the study of auxiliary Markov chains. We provide as examples general autocatalytic cores of Type I and Type III in the typology of Blokhuis et al. In a companion article, Lyapunov exponents and the behavior in the growth regime are studied quantitatively beyond the present diluted regime .

q-bio.MN↗

arXiv2019

Global fluctuations for 1D log-gas dynamics. (2) Covariance kernel and support

We consider the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations, $ dλ_t^i=\frac{1}{\sqrt{N}} dW_t^i - V'(λ_t^i) dt+ \fracβ{2N} \sum_{j\not=i} \frac{dt}{λ^i_t-λ^j_t}, \qquad i=1,\ldots,N, $ with $β>1$, sometimes called generalized Dyson's Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $β$-ensemble, with sufficiently regular convex potential $V$. The limit $N\to\infty$ is known to satisfy a mean-field Mc Kean-Vlasov equation. Fluctuations around this limit have been shown by the author to define a Gaussian process solving some explicit martingale problem written in terms of a generalized transport equation. We prove a series of results concerning either the Mc Kean-Vlasov equation for the density $ρ_t$, notably regularity results and time-evolution of the support, or the associated hydrodynamic fluctuation process, whose space-time covariance kernel we compute explicitly.

math.PR↗

arXiv2016

Dynamical invariance for random matrices

We consider a general Langevin dynamics for the one-dimensional N-particle Coulomb gas with confining potential $V$ at temperature $β$. These dynamics describe for $β=2$ the time evolution of the eigenvalues of $N\times N$ random Hermitian matrices. The equilibrium partition function -- equal to the normalization constant of the Laughlin wave function in fractional quantum Hall effect -- is known to satisfy an infinite number of constraints called Virasoro or loop constraints. We introduce here a dynamical generating function on the space of random trajectories which satisfies a large class of constraints of geometric origin. We focus in this article on a subclass induced by the invariance under the Schrödinger-Virasoro algebra.

math-ph↗

arXiv2015

PDE estimates for multi-dimensional KPZ equation

We study in this series of articles the Kardar-Parisi-Zhang (KPZ) equation $$ \partial_t h(t,x)=νΔh(t,x)+λV(|\nabla h(t,x)|) +\sqrt{D}\, η(t,x), \qquad x\in{\mathbb{R}}^d $$ in $d\ge 1$ dimensions. The forcing term $η$ in the right-hand side is a regularized white noise. The deposition rate $V$ is assumed to be isotropic and convex. Assuming $V(0)\ge 0$, one finds $V(|\nabla h|)\ltimes |\nabla h|^2$ for small gradients, yielding the equation which is most commonly used in the literature. The present article is dedicated to existence results and PDE estimates for the solution. Our results extend in a non-trivial way those previously obtained for the noiseless equation. We prove in particular a comparison principle for sub- and supersolutions of the KPZ equation in new functional spaces containing unbounded functions, implying existence and uniqueness. These new functional spaces made up of functions with "locally bounded averages", generically called ${\cal W}$-spaces thereafter, and which may be of interest for the study of parabolic equations in general, allow local or pointwise estimates. The comparison to the linear heat equation through a Cole-Hopf transform is an essential ingredient in the proofs, and our results are accordingly valid only for a function $V$ with at most quadratic growth at infinity.

math.AP↗

arXiv2015

Global existence for strong solutions of viscous Burgers equation. (1) The bounded case

We prove that the viscous Burgers equation has a globally defined smooth solution in all dimensions provided the initial condition and the forcing term are smooth and bounded together with their derivatives. Such solutions may have infinite energy. The proof does not rely on energy estimates, but on a combination of the maximum principle and quantitative Schauder estimates. We obtain precise bounds on the sup norm of the solution and its derivatives, making it plain that there is no exponential increase in time. In particular, these bounds are time-independent if the forcing term is zero. To get a classical solution, it suffices to assume that the initial condition and the forcing term have bounded derivatives up to order two.

math.AP↗

arXiv2015

Global existence and smoothness for solutions of viscous Burgers equation. (2) The unbounded case: a characteristic flow study

We show that the homogeneous viscous Burgers equation $(\partial_t-ηΔ) u(t,x)+(u\cdot\nabla)u(t,x)=0,\ (t,x)\in{\mathbb{R}}_+\times{\mathbb{R}}^d$ $(d\ge 1, η>0)$ has a globally defined smooth solution if the initial condition $u_0$ is a smooth function growing like $o(|x|)$ at infinity. The proof relies mostly on estimates of the random characteristic flow defined by a Feynman-Kac representation of the solution. Viscosity independent a priori bounds for the solution are derived from these. The regularity of the solution is then proved for fixed $η>0$ using Schauder estimates. The result extends with few modifications to initial conditions growing abnormally large in regions with small relative volume, separated by well-behaved bulk regions, provided these are stable under the characteristic flow with high probability. We provide a large family of examples for which this loose criterion may be verified by hand.

math.AP↗

arXiv2012

The Poincare algebra in the context of ageing systems: Lie structure, representations, Appell systems and coherent states

By introducing an unconventional realization of the Poincare algebra alt_1 of special relativity as conformal transformations, we show how it may occur as a dynamical symmetry algebra for ageing systems in non-equilibrium statistical physics and give some applications, such as the computation of two-time correlators. We also discuss infinite-dimensional extensions of alt_1 in this setting. Finally, we construct canonical Appell systems, coherent states and Leibniz functions for alt_1 as a tool for bosonic quantization.

math-ph↗

arXiv2012

Minkowski curvelets and wave equations

We define a new type of wavelet frame adapted to the study of wave equations, that we call Minkowski curvelets, by reference to the curvelets introduced by Candès, Demanet and Donoho. These space-time, strongly anisotropic, directional wavelets have a Fourier support which does not intersect the light-cone; their maximal size is proportional to the inverse of the distance to the light-cone. We show that the matrix of the Green kernel of the Klein-Gordon operator on Minkowski space-time has a nearly exponential off-diagonal decay in this basis.

math-ph↗

arXiv2010

A renormalized rough path over fractional Brownian motion

We construct in this article a rough path over fractional Brownian motion with arbitrary Hurst index by (i) using the Fourier normal ordering algorithm introduced in \cite{Unt-Holder} to reduce the problem to that of regularizing tree iterated integrals and (ii) applying the Bogolioubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormalization algorithm to Feynman diagrams representing tree iterated integrals.

math.PR↗

arXiv2010

Ordered forests, permutations and iterated integrals

We construct an explicit Hopf algebra isomorphism from the algebra of heap-ordered trees to that of quasi-symmetric functions, generated by formal permutations, which is a lift of the natural projection of the Connes-Kreimer algebra of decorated rooted trees onto the shuffle algebra. This isomorphism gives a universal way of lifting measure-indexed characters of the Connes-Kreimer algebra into measure-indexed characters of the shuffle algebra, already introduced in \cite{Unterberger} in the framework of rough path theory as the so-called Fourier normal ordering algorithm.

math.CO↗

arXiv2009

A Hamiltonian action of the Schrödinger-Virasoro algebra on a space of periodic time-dependent Schrödinger operators in $(1+1)$-dimensions

Let ${\cal S}^{lin}:=\{a(t)(-2\II \partial_t-\partial_r^2+V(t,r) | a\in C^{\infty}(\R/2π\Z), V\in C^{\infty}(\R/2π\Z\times\R)\}$ be the space of Schrödinger operators in $(1+1)$-dimensions with periodic time-dependent potential. The action on ${\cal S}^{lin}$ of a large infinite-dimensional reparametrization group $SV$ with Lie algebra $\sv$ \cite{RogUnt06,Unt08}, called the Schrödinger-Virasoro group and containing the Virasoro group, is proved to be Hamiltonian for a certain Poisson structure on ${\cal S}^{lin}$. More precisely, the infinitesimal action of $\sv$ appears to be part of a coadjoint action of a Lie algebra of pseudo-differential symbols, $\g$, of which $\sv$ is a quotient, while the Poisson structure is inherited from the corresponding Kirillov-Kostant-Souriau form.

math-ph↗

arXiv2009

A Lévy area by Fourier normal ordering for multidimensional fractional Brownian motion with small Hurst index

The main tool for stochastic calculus with respect to a multidimensional process $B$ with small Hölder regularity index is rough path theory. Once $B$ has been lifted to a rough path, a stochastic calculus -- as well as solutions to stochastic differential equations driven by $B$ -- follow by standard arguments. Although such a lift has been proved to exist by abstract arguments \cite{LyoVic07}, a first general, explicit construction has been proposed in \cite{Unt09,Unt09bis} under the name of Fourier normal ordering. The purpose of this short note is to convey the main ideas of the Fourier normal ordering method in the particular case of the iterated integrals of lowest order of fractional Brownian motion with arbitrary Hurst index.

math.PR↗

arXiv2009

A rough path over multidimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering

Fourier normal ordering \cite{Unt09bis} is a new algorithm to construct explicit rough paths over arbitrary Hölder-continuous multidimensional paths. We apply in this article the Fourier normal ordering ordering algorithm to the construction of an explicit rough path over multi-dimensional fractional Brownian motion $B$ with arbitrary Hurst index $α$ (in particular, for $α\le 1/4$, which was till now an open problem) by regularizing the iterated integrals of the analytic approximation of $B$ defined in \cite{Unt08}. The regularization procedure is applied to 'Fourier normal ordered' iterated integrals obtained by permuting the order of integration so that innermost integrals have highest Fourier modes. The algebraic properties of this rough path are best understood using two Hopf algebras: the Hopf algebra of decorated rooted trees \cite{ConKre98} for the multiplicative or Chen property, and the shuffle algebra for the geometric or shuffle property. The rough path lives in Gaussian chaos of integer orders and is shown to have finite moments. As well-known, the construction of a rough path is the key to defining a stochastic calculus and solve stochastic differential equations driven by $B$. The article \cite{Unt09ter} gives a quick overview of the method.

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