arXiv subjects
Publications and source records attributed to Rodrigue Lelotte.
We prove the minimisers of the Lieb-Oxford bound (if any) for a fixed (and finite) number of particles $N \geq 1$ are necessarily compactly supported, extending a result proved by E. H. Lieb and S. Oxford in the one-particle case.
The celebrated c-cyclical monotonicity property is shown to boil down to the zeroth-order optimality condition for the optimal transport problem. More precisely, we show that optimality is equivalent to the non-negativity of the linear transport cost functional on the radial cone of admissible perturbations. We then utilise this point of view to extend the c-cyclical monotonicity property to the weak optimal transport problem, for which it corresponds to the first-order optimality condition, namely to the non-negativity of the linearisation of the weak transport cost functional near the optimiser. Altogether, this sheds new light on this monotonicity concept. For both classical and weak optimal transport, we show that this property characterises (under suitable assumptions) optimal transport plans. In the classical case, we recover known results of the literature but with revisited proofs.
We provide numerical evidence for the existence of phase transitions with respect to the temperature in the one-dimensional Riesz gases with non-singular pair interaction, that is particles on the line interacting via the potential $-|r|^{-s}$, where $s \in (-1, 0)$. Our numerics hint for the existence of two distinct phase transitions whose critical temperatures depend on $s$, namely a first transition which separates between a fluid and a quasisolid phase reminiscent of the Berezinski-Kosterlitz-Thouless (BKT) transition, and a second transition below which freezing occurs and the system is in a solid phase. We determine the phase diagram with respect to $s$ and the temperature $T$, which we find to be consistent with the known (or expected) results on the 1D Coulomb gas ($s = -1$), known to be a solid at all temperature, and the Dyson log--gas ($s = 0$) which exhibits a BKT transition at $T = 1/2$ and which is believed to be a fluid at all positive temperature.
We prove a conjecture regarding the asymptotic behavior at infinity of the Kantorovich potential for the Multimarginal Optimal Transport with Coulomb and Riesz costs.
In this paper, we study the multimarginal optimal transport with Coulomb cost, also known in the physics literature as the Strictly-Correlated Electrons (SCE) functional. We prove that the dual Kantorovich potential is an electrostatic potential induced by an external charge density, which we call the dual charge. We study its properties and use it to discretize the potential in one and three space dimensions.