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Jeanne Boursier

Publications and source records attributed to Jeanne Boursier.

10 recordsLinked to original sources
arXiv2026

Free energy of the Hopfield model

We identify the limit of the free energy of the Hopfield model with independent symmetric Bernoulli patterns, for every temperature and loading ratio. A Hubbard-Stratonovich transformation reduces the problem to that of identifying the limit free energy of a bipartite spin glass with $\pm 1$ and spherical spins. One key difficulty is that the patterns, which become the coupling coefficients in the bipartite model, are Bernoulli instead of Gaussian, and the limit free energy is sensitive to the law of these coefficients. We show that the model is in fact equivalent to one involving Gaussian patterns and also subject to a finite number of Bernoulli random fields. We then solve this model by adapting the techniques recently introduced to identify the limit free energies of spin glasses with either $\pm 1$ or spherical spins.

math.PR↗
arXiv2026

Decay of correlations and thermodynamic limit for the circular Riesz gas

We study the circular Riesz gas, a one-dimensional system of particles with power-law repulsion. For the long-range gas, we prove that the correlation between two gaps at distance $d$ decays like $d^{s-2}$, up to an arbitrarily small loss in the exponent, and that this exponent is sharp. As a consequence, the microscopic point process converges as the number of particles tends to infinity, and its limit is hyperuniform. We also prove the decay of correlations and the convergence of the microscopic point process for the hypersingular gas. The proof analyzes the Helffer-Sjöstrand equation, an elliptic system in high dimension that encodes covariances. It combines the deformation of Combes and Thomas with discrete elliptic regularity estimates and a factorization around the equally spaced configuration.

math.PR↗
arXiv2026

Optimal local laws and CLT for the circular Riesz gas

We study the circular Riesz gas, a one-dimensional system of particles with long-range power-law repulsion. We establish near-optimal rigidity estimates for the gaps at all scales. We then prove a quantitative central limit theorem for linear statistics, combining Stein's method with a mean-field transport. The main difficulty is to control the fluctuations of the local error terms left by the transport, which we do using the rigidity estimates, convexity and monotonicity. In particular, a comparison principle for the Helffer--Sjöstrand equation allows us to treat test functions with low regularity, including indicators of intervals.

math.PR↗
arXiv2026

Dipole Formation in the Two-Component Plasma

We consider the two-dimensional two-component plasma, or Coulomb gas, consisting of $N$ positive and $N$ negative charges with logarithmic interaction. We introduce a suitable regularization of the interaction by smearing the charges over a small length scale $λ$, which allows us to give meaning to the system in the continuum at any temperature. We provide an expansion of the free energy for each inverse temperature $β\ge2$, as $N \to \infty$ and $λ\to 0$. Doing so allows us to show that, for $β\geq 2$, the charges (for the most part) pair into neutral dipoles of very small size as $λ\to 0$. This complements the prior work of Leblé, Zeitouni, and the second author, which proved that this does not happen for $β< 2$, thereby implying a transition at $β= 2$ from free charges to dipole pairs. Moreover, we obtain an estimate on the size of linear statistics. The description in terms of dipoles is made via a decomposition into nearest-neighbor graphs of the point configurations, à la Gunson-Panta. This is combined with new energy estimates obtained via an electric reformulation of the interaction energy and a ball-growth method, which are expressed in terms of the nearest-neighbor graph distances only. In this way, the model is compared to a reduced nearest-neighbor interaction model, by showing the relative smallness of the dipole-dipole interactions.

math-ph↗
arXiv2026

Free energy of non-convex multi-species Ising spin glasses with external fields

We identify the limit free energy of mean-field multi-species spin glasses with +/-1 spins, allowing for the presence of external fields, and without assuming any convexity property of the covariance function of the Hamiltonian. The limit is given by an "uninverted" variational formula over Brownian martingales. The proof consists in first identifying the limit free energy of configurations with a fixed magnetization. Once the magnetization is fixed, we can show the convergence of the enriched free energy to the solution of a Hamilton-Jacobi equation. The magnetization constraint is also used to show the convexity of the initial condition in this equation, and this yields a variational representation of the solution, as a Hopf formula. Convex duality then turns this formula into an optimization over martingales whose initial values are the prescribed magnetizations.

math.PR↗
arXiv2026

Large deviations for the maximum of the generalized TAP free energy

We study large deviations of the maximum of the generalized TAP free energy introduced by Chen, Panchenko, and Subag for the Ising mixed $p$-spin model. The upper bound uses a new Guerra-type interpolation that may also be useful for other spin-glass models. The lower bound follows the strategy of Huang and Sellke. We show that supersymmetric critical points have no competitor at positive overlap on their own sphere. As such, they form a spherical code, which turns the annealed count into a lower bound on the probability of existence. This identifies the supersymmetric formula proposed by physicists with the large-deviation exponent for the existence of TAP maxima, rather than with the ordinary annealed complexity. The resulting rate function is the Legendre transform of a constrained Parisi value in the bottom-atom mass. The same argument, iterated in bands above TAP ancestors, would give a constructive proof of the Parisi formula for the Ising model under a technical strict stability assumption.

math.PR↗
arXiv2026

The Legendre structure of the TAP complexity for the Ising spin glass

We study the complexity of the Thouless-Anderson-Palmer (TAP) free energy for Ising spin glasses with a general mixed p-spin covariance, working with the generalized TAP functional of Chen, Panchenko, and Subag. We formulate three conjectures about the complexity (i.e. number of critical points). First, the annealed complexity is given by the Legendre transform of a variational functional constructed from the Parisi formula subject to a constraint on the overlap mass at zero, thereby establishing a precise link between the enumeration of TAP states and the large-deviation rate function of the partition function. Second, the quenched complexity is governed by the Legendre transform of a closely related functional in which the mass up to -- but not including -- the supremum of the support is constrained. Third, TAP states at any non-equilibrium free-energy level are organized into an ultrametric hierarchy, with ancestor states at other levels appearing only in subexponential number. Using a Kac-Rice computation combined with a supersymmetric ansatz, we establish a lower bound on the annealed complexity that matches the prediction of the first conjecture. We further extend the analysis to a conditional setting in which a hierarchical "skeleton" of ancestors is prescribed, providing additional evidence in support of the second and third conjectures.

math.PR↗
arXiv2026

Multipole and Berezinskii-Kosterlitz-Thouless Transitions in the Two-component Plasma

We study the two-dimensional two-component Coulomb gas in the canonical ensemble and at inverse temperature $β>2$. In this regime, the partition function diverges and the interaction needs to be cut off at a length scale $λ\in (0,1)$. Particles of opposite charges tend to pair into dipoles of length scale comparable to $λ$, which themselves can aggregate into multipoles. Despite the slow decay of dipole--dipole interactions, we construct a convergent cluster expansion around a hierarchical reference model that retains only intra-multipole interactions. This yields a large deviations result for the number of $2p$-poles as well as a sharp free energy expansion as $N\to\infty$ and $λ\to0$ with three contributions: (i) the free energy of $N$ independent dipoles, (ii) a perturbative correction, and (iii) the contribution of a non-dilute subsystem. The perturbative term has two equivalent characterizations: (a) a convergent Mayer series obtained by expanding around an i.i.d.\ dipole model; and (b) a variational formula as the minimum of a large-deviation rate function for the empirical counts of $2p$-poles. The Mayer coefficients exhibit transitions at $β_p=4-\tfrac{2}{p}$, that accumulate at $β=4$, which corresponds to the Berezinskii-Kosterlitz-Thouless transition in the low-dipole-density limit. At $β=β_p$ the $p$-dipole cluster integrals switch from non-integrable to integrable tails. The non-dilute system corresponds to the contribution of large dipoles: we exhibit a new critical length scale $R_{β, λ}$ which transitions from $λ^{-(β-2)/(4-β)}$ to $+\infty$ as $β$ crosses the critical inverse temperature $β=4$, and which can be interpreted as the maximal scale such that the dipoles of that scale form a dilute set.

math-ph↗
arXiv2022

Universal cutoff for Dyson Ornstein Uhlenbeck process

We study the Dyson-Ornstein-Uhlenbeck diffusion process, an evolving gas of interacting particles. Its invariant law is the beta Hermite ensemble of random matrix theory, a non-product log-concave distribution. We explore the convergence to equilibrium of this process for various distances or divergences, including total variation, relative entropy, and transportation cost. When the number of particles is sent to infinity, we show that a cutoff phenomenon occurs: the distance to equilibrium vanishes abruptly at a critical time. A remarkable feature is that this critical time is independent of the parameter beta that controls the strength of the interaction, in particular the result is identical in the non-interacting case, which is nothing but the Ornstein-Uhlenbeck process. We also provide a complete analysis of the non-interacting case that reveals some new phenomena. Our work relies among other ingredients on convexity and functional inequalities, exact solvability, exact Gaussian formulas, coupling arguments, stochastic calculus, variational formulas and contraction properties. This work leads, beyond the specific process that we study, to questions on the high-dimensional analysis of heat kernels of curved diffusions.

math.PR↗