On the log-Sobolev constant of log-concave measures

It is well known that a log-Sobolev inequality implies sub-gaussian decay of the tails. In the spirit of the KLS conjecture, we investigate whether this implication can be reversed under a log-concavity assumption. In the general setting, we improve on a result of Bobkov, establishing the best dimension dependent bound on the log-Sobolev constant of subgaussian log-concave measures, and we investigate some special cases

Positive solutions for large random linear systems

Consider a large linear system where $A_n$ is a $n\times n$ matrix with independent real standard Gaussian entries, $\boldsymbol{1}_n$ is a $n\times 1$ vector of ones and with unknown the $n\times 1$ vector $\boldsymbol{x}_n$ satisfying$$\boldsymbol{x}_n = \boldsymbol{1}_n +\frac 1{\alpha_n\sqrt{n}} A_n \boldsymbol{x}_n\, .$$We investigate the (componentwise) positivity of the solution $\boldsymbol{x}_n$ depending on the scaling factor $\alpha_n$ as the dimension $n$ goes to $\infty$. We prove that there is a sharp phase transition at the threshold $\alpha^*_n =\sqrt{2\log n}$: below the threshold ($\alpha_n\ll \sqrt{2\log n}$), $\boldsymbol{x}_n$ has negative components with probability tending to 1 while above ($\alpha_n\gg \sqrt{2\log n}$), all the vector's components are eventually positive with probability tending to 1. At the critical scaling $\alpha^*_n$, we provide a heuristics to evaluate the probability that $\boldsymbol{x}_n$ is positive.Such linear systems arise as solutions at equilibrium of large Lotka-Volterra systems of differential equations, widely used to describe large biological communities with interactions such as foodwebs for instance. In the domaine of positivity of the solution $\boldsymbol{x}_n$, that is when $\alpha_n\gg \sqrt{2\log n}$, we establish that the Lotka-Volterra system of differential equations whose solution at equilibrium is precisely \x_n is stable in the sense that its jacobian$${\mathcal J}(\boldsymbol{x}_n) = \mathrm{diag}(\boldsymbol{x}_n)\left(-I_n + \frac {A_n}{\alpha_n\sqrt{n}}\right)$$has all its eigenvalues with negative real part with probability tending to one.Our results shed a new light and complement the understanding of feasibility and stability issues for large biological communities with interaction

Méthodes stochastiques en convexité

This thesis deals with high-dimensionnal phenomena arising under convexity assumptions. In a first part, we study the behavior of the entropy and information with respect to convolutions of log-concave vectors. Then, using stochastic localization, a very recent technique which led to an almost resolution of the KLS conjecture, we establish new results regarding the concentration fucntion of log-concave probabilities, and their log-Sobolev constant. Finally, the last chapter is devoted to the study of large random linear systems, for which a cut-off phenomenon is established.Cette thèse s'inscrit dans le cadre des probabilités en grande dimension, en particulier sous hypothèse de convexité. Dans une première partie, on étudie le comportement des l'entropie et de l'information de Fisher vis à vis des convolutions de vecteurs log-concave. Ensuite, à l'aide de la localisation stochastique, une technique récente qui a notamment servi à la quasi résolution de la conjecture KLS, nous établissons des résultats nouveaux sur la fonction de concentration des mesures log-concave, et leur constante de log-sobolev. La dernière partie est consacrée à l'étude de grands systèmes linéaires aléatoires pour lesquels un phénomène de type cut-off est démontré

Positive solutions for large random linear systems

International audienceConsider a large linear system where $A_n$ is a $n\times n$ matrix with independent real standard Gaussian entries, $\boldsymbol{1}_n$ is a $n\times 1$ vector of ones and with unknown the $n\times 1$ vector $\boldsymbol{x}_n$ satisfying$$\boldsymbol{x}_n = \boldsymbol{1}_n +\frac 1{\alpha_n\sqrt{n}} A_n \boldsymbol{x}_n\, .$$We investigate the (componentwise) positivity of the solution $\boldsymbol{x}_n$ depending on the scaling factor $\alpha_n$ as the dimension $n$ goes to $\infty$. We prove that there is a sharp phase transition at the threshold $\alpha^*_n =\sqrt{2\log n}$: below the threshold ($\alpha_n\ll \sqrt{2\log n}$), $\boldsymbol{x}_n$ has negative components with probability tending to 1 while above ($\alpha_n\gg \sqrt{2\log n}$), all the vector's components are eventually positive with probability tending to 1. At the critical scaling $\alpha^*_n$, we provide a heuristics to evaluate the probability that $\boldsymbol{x}_n$ is positive.Such linear systems arise as solutions at equilibrium of large Lotka-Volterra systems of differential equations, widely used to describe large biological communities with interactions such as foodwebs for instance. In the domaine of positivity of the solution $\boldsymbol{x}_n$, that is when $\alpha_n\gg \sqrt{2\log n}$, we establish that the Lotka-Volterra system of differential equations whose solution at equilibrium is precisely \x_n is stable in the sense that its jacobian$${\mathcal J}(\boldsymbol{x}_n) = \mathrm{diag}(\boldsymbol{x}_n)\left(-I_n + \frac {A_n}{\alpha_n\sqrt{n}}\right)$$has all its eigenvalues with negative real part with probability tending to one.Our results shed a new light and complement the understanding of feasibility and stability issues for large biological communities with interaction

Positive Solutions for Large Random Linear Systems

International audienceConsider a large linear system with random underlying matrix: xn = 1n + 1/(αn √βn) Mn xn, where xn is the unknown, 1n is a vector of ones, Mn is a random matrix and αn, βn are scaling parameters to be specified. We investigate the componentwise positivity of the solution x n depending on the scaling factors, as the dimensions of the system grow to infinity.We consider 2 models of interest: The case where matrix Mn has independent and identically distributed standard Gaussian random variables, and a sparse case with a growing number of vanishing entries.In each case, there exists a phase transition for the scaling parameters below which there is no positive solution to the system with growing probability and above which there is a positive solution with growing probability.These questions arise from feasibility and stability issues for large biological communities with interactions

Quantifying pesticide-contaminated sediment sources in tropical coastal environments (Galion Bay, French West Indies)

International audiencePurpose Over the last 60 years, intensification of soil cultivation led to an acceleration of soil erosion and sediment delivery to river systems. In Martinique, this acceleration has led to the remobilization of a toxic insecticide (i.e. chlordecone) used in the 1970s-1990s to control banana weevil. A previous study attributed this accelerated remobilization to the application of glyphosate in plantations from the 1990s onwards. To further unambiguously confirm this link, the identification of soil erosion sources supplied to coastal sediment is essential. Methods Accordingly, sediment fingerprinting tools were adapted and applied to a coastal sediment core collected in the Galion Bay. Potential source samples (n=37) were collected across the drainage area. Along with the coastal sediment core layers, these samples were analysed for potential tracing properties. The optimal suite of tracers was then selected and introduced into an un-mixing model to quantify their contributions to coastal sediment. Results Results showed that subsoil (i.e. soil layer < 30 cm depth) and banana plantation soil surface supply the major sources of sediment (49-78% and 12-36%, respectively) to the Galion Bay and that their contributions increased since 2000, in line with chlordecone and glyphosate fluxes. Conclusion This evolution may be attributed to the higher sensitivity of banana plantations to erosion that may have been enhanced by the glyphosate application leaving the soil uncovered with vegetation and to the contamination of both topsoil and deep soil layers (< 30 cm) layers with chlordecone due to its vertical transfer along the soil profile and its redistribution across hillslope

Was the 137Cs contained in Saharan dust deposited across Europe in March 2022 emitted by French nuclear tests in Algeria?

Air masses loaded with mineral dust and originating from the Sahara arrive frequently in Europe, which has multiple impacts on global and regional cycles. However, the occurrence of these processes may further accelerate in the future in response to climate change, and more knowledge is therefore required on the characteristics of the particles transported during these massive dust transport and deposition episodes. Furthermore, questions arise regarding the content of this dust in radionuclides, in relationship with the atmospheric nuclear bomb testing conducted around the world between the 1950s and the 1970s in general, and those tests conducted by France in the Sahara in the early 1960s in particular. The Saharan dust episode that took place from 13 to 16 March 2022 led to the occurrence of dense dust deposition across multiple European countries, which raised concerns among the population regarding the potential radioactivity content of this dust. To address this question with a representative sample set, a participative science campaign to collect dust across Europe was launched on Twitter on 17 March 2022. Thanks to this initiative, 110 dust samples could be collected along a transect from Southern Spain to Austria. This unique sample bank was regrouped at University Paris-Saclay, France, to conduct a set of physico-chemical analyses on a selection or on the totality of these dust samples including particle size, colourimetry, mineralogy and fallout radionuclides. Backward trajectories of air masses that have led to these deposits were calculated, and this analysis confirm their potential origin from Algeria. 137Cs was detected in all dust samples, with variable activity concentrations. A strong relationship was found between the particle size of the analysed particles and the 137Cs activity concentrations, which is consistent with the literature on this topic. Particle size was found to decrease with increasing distances from the source. The colour and mineralogy analyses demonstrated that the dust collected in Austria showed different properties than those samples collected in Spain, France, Luxembourg and Germany, which likely indicates that this material did not fully consist of Saharan dust deposited during the March 2022 episode. Accordingly, the following interpretations did not take the properties of Austrian dust into account. The mineralogical analyses confirmed the potential origin of the dust from the Maghreb region, including a vast area in Southern Morocco and Southern Algeria. In contrast, the analysis of plutonium isotopic ratios (240Pu/239Pu) and 137Cs/239+240Pu activity ratios, which provide diagnosis tools to investigate the source of artificial radionuclides, in a selection of dust samples collected between Southern Spain and Luxembourg showed that the dust signature was consistent with that of the global fallout largely dominated by the nuclear tests conducted by the USA and the Soviet Union. The 137Cs contained in the dust transported and deposited during this episode was therefore very likely not associated with the French nuclear tests conducted in the early 1960s in Sahara. In the future, elemental geochemistry analyses will provide additional information on their source provenance. All results will also be published in open-access database and disseminated to the public