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Matthew Rosenzweig

Publications and source records attributed to Matthew Rosenzweig.

At least 19 recordsLinked to original sources
arXiv2026

Sharp mean-field estimates for diffusive log/Riesz gases in the Hilbert--Schmidt regime

We study fixed-temperature logarithmic and Riesz gases after subtracting the leading mean-field contribution relative to a prescribed background law. For repulsive interactions in the Hilbert--Schmidt regime, we prove $N$-uniform bounds and quantitative convergence of the resulting modulated partition function to a normalization expressed by the Carleman--Fredholm determinant of the centered interaction operator. We show that the Hilbert--Schmidt threshold is sharp and obtain explicit lower bounds on the rate of divergence at and above it; these rates are expected to be nonoptimal. The proof combines positive-definite truncations and a low/high-frequency decomposition with exponential inequalities and Gaussian-chaos asymptotics for canonical degree-two $U$-statistics. As consequences, we establish entropic commutator estimates with the sharp $O(N^{-1})$ additive scale in the modulated-free-energy method, a static joint linear-statistics central limit theorem, and a dynamical central limit theorem for joint linear statistics at finitely many times. This extends the logarithmic partition-function estimates of the first two authors to the full Riesz Hilbert--Schmidt range and identifies the limiting determinant normalization. For the attractive logarithmic interaction at sufficiently small inverse temperature, we also prove analogous results.

math.PR↗
arXiv2026

Uniform logarithmic Sobolev inequalities for the 2D Coulomb gas at the diffusive temperature scale

For $N\ge2$ and $β>0$, consider the canonical 2D Coulomb gas ensemble \begin{equation} \mathrm{d}\mathbb{P}_{N,β}(Z) =\mathsf{Z}_{N,β}^{-1}e^{-β|Z|^2/2} \prod_{i 0$, this Gibbs measure satisfies a logarithmic Sobolev inequality (LSI) on the closed Dirichlet-form domain generated by collision-free compactly supported smooth functions, with a positive constant independent of $N$. The LSI constant may be chosen uniformly when $β$ ranges over a compact subset of $(0,\infty)$. A second main result establishes Poincaré and logarithmic Sobolev inequalities for planar Gaussian measures weighted by finite products of positive powers of distances to points, with constants depending only on the total variance of the underlying Gaussian measure and the total exponent, not on the number or locations of the points. To prove the full labeled Poincaré inequality, we decompose variance into permutation-invariant and label-dependent parts, controlled respectively by a weighted $\bar\partial$ estimate and by relative-coordinate conditioning with random transpositions; separately, one-site logarithmic Sobolev inequalities, a conditional entropy inequality, and a repulsive partition estimate give a defective logarithmic Sobolev inequality. Rothaus tightening combines the resulting full labeled Poincaré inequality with the defective logarithmic Sobolev inequality. No uniformity as $β\to0$ or $β\to\infty$ is asserted.

math.PR↗
arXiv2026

Fixed-particle-number optimizers for the Lieb--Oxford inequality

Let $\mathsf{d}\geq1$, $0<\mathsf{s}<\mathsf{d}$, and $N\geq1$. We prove that the optimal fixed-particle-number constant $Λ_N(\mathsf{s},\mathsf{d})$ in the Riesz Lieb--Oxford inequality is attained and that these constants are strictly increasing in $N$. The proof combines grand-canonical concentration--compactness with a strict one-particle extension. After recentering, a limiting plan arising from a maximizing sequence may assign positive probability to several particle numbers and hence be grand-canonical. A strict $N$-particle completion excludes this case, while the inequalities $Λ_N>Λ_k$ for $k Λ_N(\mathsf{s},\mathsf{d})$. Together, these implications close an induction beginning at $N=1$.

math.AP↗
arXiv2026

Wasserstein gradient flows of Maximum Mean Discrepancy with energy kernels

We study the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) generated by the nonsmooth energy kernels $K(z)=-|z|^q$, $0 0$, we prove global well-posedness on $\mathbb{R}^d$ for probability densities in subcritical $L^p$ spaces, with targets in the same integrability class and with finite moments. We also include the one-dimensional Coulomb endpoint $d=q=1$. For the associated $N$-particle system, we prove global noncollision and fixed-$N$ convergence to the collision-free critical set, a particle-to-continuum criticality principle, and a modulated-energy mean-field estimate that yields convergence of the particle dynamics to the continuum flow as $N\to\infty$ on every finite time interval. We also construct collision-free saddle equilibria, showing that deterministic particle trajectories need not approach global empirical minimizers. For $1\le q<2$, every continuum solution in our class has a narrowly relatively compact orbit, every $ω$-limit point is Lagrangian critical, and the orbit approaches the Lagrangian critical set. For $0<q<1$, the same conclusions hold under uniform-in-time moment and subcritical $L^p$ bounds. We prove that an absolutely continuous Lagrangian critical point equals the target when the source and target have finite moments of order $q$, except when $0<q<1$ and $d\in\{1,3\}$. Under the preceding uniform bounds, rigidity gives convergence of the continuum flow to the target throughout the rigid part of the well-posedness range. Finally, we show that no initial-data-independent multiplicative MMD decay modulus exists on $\mathbb{R}^d$, and that global Polyak--Łojasiewicz inequalities fail in several whole-space and periodic Riesz/Coulomb regimes.

math.AP↗
arXiv2026

Wasserstein gradient flows for Coulomb discrepancies

We study the long-time behavior of the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) between a probability measure $ρ$ and a target measure $μ$, where the underlying kernel is given by a Coulomb potential. For $L^\infty$ target densities $μ$, we establish the existence of global weak solutions starting from arbitrary Borel probability measures and prove that the density $ρ_t$ belongs to $L^\infty$ for any $t>0$. We also show that the Hölder norm can grow exponentially in time. On the flat torus ${\mathbb{T}}^\mathsf{d}$, we prove a global metric PL inequality for every finite-Coulomb-energy source and nearly uniform target. For general bounded, uniformly positive targets, we prove exponential decay of the squared MMD without requiring a lower bound on the initial data, using a defective PL inequality. We also prove that the usual PL inequality may fail when the target vanishes only at one point and that, when $\mathsf{d}\ge2$, no PL constant can hold uniformly over all targets satisfying a prescribed lower bound. On ${\mathbb{R}}^\mathsf{d}$, for $\mathsf{d}\ge2$, under radial symmetry, source-support inclusion, and target-positivity assumptions, we establish a PL inequality and exponential convergence. On the unrestricted whole-space class, neither a multiplicative squared-MMD decay modulus uniform over the initial datum nor a global PL inequality can hold. Finally, in every dimension and in both spatial settings, we prove that every Lagrangian critical point coincides with the target when $(ρ-μ)^+$ is absolutely continuous. In dimension two, the energy supplies uniform tightness. This implies that if our constructed solutions have finite energy at some positive time, then they converge to the target narrowly and strongly in negative-order Sobolev spaces.

math.AP↗
arXiv2026

Phase transitions for the noisy transformer model in arbitrary dimension

We study the McKean--Vlasov free energy on the unit sphere associated with the unnormalized self-attention (USA) model for noisy transformer dynamics. We prove a sharp global-minimizer dichotomy in every dimension $d\ge2$. There is a unique $β_*^{(d)}>0$ such that \begin{equation*} \frac{I_{d/2+1}(β_*^{(d)})}{I_{d/2}(β_*^{(d)})}=\frac1d, \end{equation*} where $I_ν$ is the modified Bessel function of the first kind. For $0<β\le β_*^{(d)}$, the uniform density remains the unique global minimizer up to the linear-stability threshold \begin{equation*} K_\#^{(d)}(β)=\frac{β^{d/2}}{2^{d/2}Γ(d/2)I_{d/2}(β)}, \end{equation*} and the phase transition is continuous. For $β>β_*^{(d)}$, the uniform density is not globally minimizing at $K_\#^{(d)}(β)$, so the critical coupling satisfies $K_c<K_\#^{(d)}(β)$ and the transition is discontinuous. This result generalizes the authors' recent $d=2$ work arXiv:2604.16288 to arbitrary dimension. The proof uses the sharp Beckner--Onofri/logarithmic Hardy-Littlewood-Sobolev (HLS) inequality on the sphere, together with a Funk--Hecke/Bessel coefficient computation and a degree-two quartic obstruction.

math.AP↗
arXiv2026

Phase transitions in Doi-Onsager, Noisy Transformer, and other multimodal models

We study phase transitions for repulsive-attractive mean-field free energies on the circle. For a $\frac{1}{n+1}$-periodic interaction whose Fourier coefficients satisfy a certain decay condition, we prove that the critical coupling strength $K_c$ coincides with the linear stability threshold $K_\#$ of the uniform distribution and that the phase transition is continuous, in the sense that the uniform distribution is the unique global minimizer at criticality. The proof is based on a sharp coercivity estimate for the free energy obtained from the constrained Lebedev--Milin inequality. We apply this result to three motivating models for which the exact value of the phase transition and its (dis)continuity in terms of the model parameters was not fully known. For the two-dimensional Doi--Onsager model $W(θ)=-|\sin(2πθ)|$, we prove that the phase transition is continuous at $K_c=K_\#=3π/4$. For the noisy transformer model $W_β(θ)=(e^{β\cos(2πθ)}-1)/β$, we identify the sharp threshold $β_*$ such that $K_c(β) = K_\#(β)$ and the phase transition is continuous for $β\leq β_*$, while $K_c(β) β_*$. We also obtain the corresponding sharp dichotomy for the noisy Hegselmann--Krause model $W_{R}(θ) = (R-2π|θ|)_{+}^2$ .

math.AP↗
arXiv2026

Commutators, mean-field, and supercritical mean-field limits for Coulomb/Riesz gases

This paper is intended as a companion to the author's talk "Commutator estimates and mean-field limits for Coulomb/Riesz gases" at the 2025 Journées équations aux dérivées partielles in Aussois. The goal is to provide a concise, accessible account of sharp commutator estimates recently obtained for modulated energies associated to Coulomb/Riesz interactions and how these estimates lead to optimal results for mean-field and supercritical mean-field limits of Coulomb/Riesz gas dynamics via the modulated-energy method. The exposition centers on the works arXiv:2408.14642, arXiv:2407.15650 with Serfaty and arXiv:2511.13461, arXiv:2601.02326 with Hess-Childs and Serfaty.

math.AP↗
arXiv2026

Phase transitions and linear stability for the mean-field Kuramoto-Daido model

We consider the mean-field noisy Kuramoto-Daido model, which is a McKean-Vlasov equation on the circle with bimodal interaction $W(θ)=\cosθ+m\cos2θ$ for $m\ge 0$ and interaction strength $K$, generalizing the celebrated noisy Kuramoto model corresponding to $m=0$. Our first contribution is to characterize the phase transition threshold $K_{c}$ by comparing it to the linear stability threshold $K_\# = \min (1, m^{-1})$ of the uniform distribution. When $m \leq 1/2,$ $K_{c}=1$, coinciding with that of the Kuramoto model. On the other hand, for $m \geq 2$, we show $K_c= m^{-1}$. We also classify the regimes in which the phase transition is continuous or discontinuous. Our second contribution is to analyze the linear stability of a global minimizer $q$ (the ``ordered phase'') of the mean-field free energy in the supercritical regime $K>1$. This stationary solution of the Kuramoto-Daido equation is unique up to translation invariance and distinct from the uniform distribution (the ``disordered phase''). Our approach extends the Dirichlet form method of Bertini et al. from the unimodal to bimodal setting. In particular, for $m \leq 1.590 \times 10^{-4}$ and $K>1$, we show an explicit lower bound on the spectral gap of the linearized McKean-Vlasov operator at $q$. To our knowledge, this is the first rigorous stability analysis for this class of models with bimodal interactions.

math.AP↗
arXiv2026

Another look at regularity in transport-commutator estimates

We are interested in how regular a transport velocity field must be in order to control Riesz-type commutators. Estimates for these commutators play a central role in the analysis of the mean-field limit and fluctuations for systems of particles with pairwise Riesz interactions, which we start by reviewing. Our first new result shows that the usual $L^\infty$ assumption on the gradient of the velocity field cannot, in general, be relaxed to a BMO assumption. We construct counterexamples in all dimensions and all Riesz singularities $-2< s<d$, except for the one-dimensional logarithmic endpoint $s=0$. At this exceptional endpoint, such a relaxation is possible, a fact related to the classical Coifman-Rochberg-Weiss commutator bound for the Hilbert transform. Our second result identifies a trade-off between the singularity of the interaction potential and the required regularity of the velocity field. Roughly speaking, smoother (less singular) interactions require stronger velocity control if one wants a commutator estimate in the natural energy seminorm determined by the potential. We formulate this principle for a broad class of potentials and show that, in the sub-Coulomb Riesz regime, the velocity regularity appearing in the known commutator inequality is sharp. Despite these negative findings, we show as our third result that a defective commutator estimate holds for almost-Lipschitz transport fields. Such a defective estimate, which is a consequence of the celebrated Brezis-Wainger-Hansson inequality, allows us to prove rates of convergence when the mean-field density belongs to the scaling-critical Sobolev space.

math.AP↗
arXiv2025

A sharp commutator estimate for all Riesz modulated energies

We prove a functional inequality in any dimension controlling the derivative along a transport of the Riesz modulated energy in terms of the modulated energy itself. This modulated energy was introduced by the third author and collaborators in the study of mean-field limits and statistical mechanics of Coulomb/Riesz gases, where this control is an essential ingredient. Previous work of the last two authors and Q.H. Nguyen arXiv:2107.02592 showed a similar functional inequality but with an additive $N$-dependent error (where $N$ is the number of particles, $\mathsf{d}$ the dimension, and $\mathsf{s}$ the inverse power of the Riesz potential) which was not sharp. In this paper, we obtain the optimal $N^{\frac{\mathsf{s}}{\mathsf{d}}-1}$ error, for all cases, including the sub-Coulomb case. Our method is conceptually simple and, like previous work, relies on the observation that the derivative along a transport of the modulated energy is the quadratic form of a commutator. Through a new potential truncation scheme based on a wavelet-type representation of the Riesz potential to handle its singularity, the proof reduces to averaging over a family of Kato-Ponce type estimates. The commutator estimate has applications to sharp rates of convergence for mean-field limits, quasi-neutral limits, and central limit theorems for the fluctuations of Coulomb/Riesz gases both at and out of thermal equilibrium. In particular, we show here for $\mathsf{s}<\mathsf{d}-2$ the expected $N^{\frac{\mathsf{s}}{\mathsf{d}}-1}$-rate in the modulated energy distance for the mean-field convergence of first-order Hamiltonian and gradient flows. This complements the recent work arXiv:2407.15650 on the optimal rate for the (super-)Coulomb case $\mathsf{d}-2\le \mathsf{s}<\mathsf{d}$ and therefore resolves the entire potential Riesz case.

math.AP↗
arXiv2025

Sharp commutator estimates of all order for Coulomb and Riesz modulated energies

We prove functional inequalities in any dimension controlling the iterated derivatives along a transport of the Coulomb or super-Coulomb Riesz modulated energy in terms of the modulated energy itself. This modulated energy was introduced by the second author and collaborators in the study of mean-field limits and statistical mechanics of Coulomb/Riesz gases, where control of such derivatives by the energy itself is an essential ingredient. In this paper, we extend and improve such functional inequalities, proving estimates which are now sharp in their additive error term, in their density dependence, valid at arbitrary order of differentiation, and localizable to the support of the transport. Our method relies on the observation that these iterated derivatives are the quadratic form of a commutator. Taking advantage of the Riesz nature of the interaction, we identify these commutators as solutions to a degenerate elliptic equation with a right-hand side exhibiting a recursive structure in terms of lower-order commutators and develop a local regularity theory for the commutators, which may be of independent interest. These estimates have applications to obtaining sharp rates of convergence for mean-field limits, quasi-neutral limits, and in proving central limit theorems for the fluctuations of Coulomb/Riesz gases. In particular, we show here the expected $N^{\frac{\mathsf{s}}{\mathsf{d}}-1}$-rate in the modulated energy distance for the mean-field convergence of first-order Hamiltonian and gradient flows.

math.AP↗
arXiv2024

The Lake equation as a supercritical mean-field limit

We study so-called supercritical mean-field limits of systems of trapped particles moving according to Newton's second law with either Coulomb/super-Coulomb or regular interactions, from which we derive a $\mathsf{d}$-dimensional generalization of the Lake equation, which coincides with the incompressible Euler equation in the simplest setting, for monokinetic data. This supercritical mean-field limit may also be interpreted as a combined mean-field and quasineutral limit, and our assumptions on the rates of these respective limits are shown to be optimal. Our work provides a mathematical basis for the universality of the Lake equation in this scaling limit -- a new observation -- in the sense that the dependence on the interaction and confinement is only through the limiting spatial density of the particles. Our proof is based on a modulated-energy method and takes advantage of regularity theory for the obstacle problem for the fractional Laplacian.

math.AP↗
arXiv2024

Relative entropy and modulated free energy without confinement via self-similar transformation

This note extends the modulated entropy and free energy methods for proving mean-field limits/propagation of chaos to the whole space without any confining potential, in contrast to previous work limited to the torus or requiring confinement in the whole space, for all log/Riesz flows. Our novel idea is a scale transformation, sometimes called self-similar coordinates in the PDE literature, which converts the problem to one with a quadratic confining potential, up to a time-dependent renormalization of the interaction potential. In these self-similar coordinates, one can then establish a Grönwall relation for the relative entropy or modulated free energy, conditional on bounds for the Hessian of the mean-field log density. This generalizes recent work of Feng-Wang arXiv:2310.05156, which extended the Jabin-Wang relative entropy method to the whole space for the viscous vortex model. Moreover, in contrast to previous work, our approach allows to obtain uniform-in-time propagation of chaos and even polynomial-in-time generation of chaos in the whole space without confinement, provided one has suitable decay estimates for the mean-field log density. The desired regularity bounds and decay estimates are the subject of a companion paper.

math.AP↗
arXiv2023

The attractive log gas: stability, uniqueness, and propagation of chaos

We consider overdamped Langevin dynamics for the attractive log gas on the torus $\mathbb{T}^\mathsf{d}$, for $\mathsf{d}\geq 1$. In dimension $\mathsf{d}=2$, this model coincides with a periodic version of the parabolic-elliptic Patlak-Keller-Segel model of chemotaxis. The attractive log gas (for our choice of units) is well-known to have a critical inverse temperature $β_{\mathrm{c}}={2\mathsf{d}}$ corresponding to when the free energy is bounded from below. Moreover, it is well-known that the uniform distribution is always a stationary state regardless of the temperature. We identify another temperature threshold $β_{\mathrm{s}}$ sharply corresponding to the nonlinear stability of the uniform distribution. We show that for $β>β_{\mathrm{s}}$, the uniform distribution does not minimize the free energy and moreover is nonlinearly unstable, while for $β<β_{\mathrm{s}}$, it is stable. We also show that there exists $β_{\mathrm{u}}$ for which uniqueness of equilibria holds for $β<β_{\mathrm{u}}$. We establish a uniform-in-time rate for entropic propagation of chaos for a range of $β<β_{\mathrm{s}}$. To our knowledge, this is the first such result for singular attractive interactions and affirmatively answers a question of Bresch et al. arXiv:2011.08022. The proof of the convergence is through the modulated free energy method, relying on a modulated logarithmic Hardy-Littlewood-Sobolev (mLHLS) inequality. Unlike Bresch et al., we show that such an inequality holds without truncation of the potential -- the avoidance of the truncation being essential to a uniform-in-time result -- for sufficiently small $β$ and provide a counterexample to the mLHLS inequality when $β>β_{\mathrm{s}}$. As a byproduct, we show that it is impossible to have a uniform-in-time rate of propagation of chaos if $β>β_{\mathrm{s}}$.

math.AP↗
arXiv2023

Modulated logarithmic Sobolev inequalities and generation of chaos

We consider mean-field limits for overdamped Langevin dynamics of $N$ particles with possibly singular interactions. It has been shown that a modulated free energy method can be used to prove the mean-field convergence or propagation of chaos for a certain class of interactions, including Riesz kernels. We show here that generation of chaos, i.e. exponential-in-time convergence to a tensorized (or iid) state starting from a nontensorized one, can be deduced from the modulated free energy method provided a uniform-in-$N$ "modulated logarithmic Sobolev inequality" holds. Proving such an inequality is a question of independent interest, which is generally difficult. As an illustration, we show that uniform modulated logarithmic Sobolev inequalities can be proven for a class of situations in one dimension.

math.PR↗
arXiv2023

Trend to equilibrium for flows with random diffusion

Motivated by the possibility of noise to cure equations of finite-time blowup, recent work arXiv:2109.09892 by the second and third named authors showed that with quantifiable high probability, random diffusion restores global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as the Patlak-Keller-Segel equation. A question left open is the asymptotic behavior of the solutions, in particular, whether they converge to a steady state. We answer this question by showing that the solutions from arXiv:2109.09892 in the periodic setting converge in Gevrey norm exponentially fast to the uniform distribution as time $t\rightarrow\infty$.

math.AP↗
arXiv2023

Sharp uniform-in-time mean-field convergence for singular periodic Riesz flows

We consider conservative and gradient flows for $N$-particle Riesz energies with mean-field scaling on the torus $\mathbb{T}^d$, for $d\geq 1$, and with thermal noise of McKean-Vlasov type. We prove global well-posedness and relaxation to equilibrium rates for the limiting PDE. Combining these relaxation rates with the modulated free energy of Bresch et al. and recent sharp functional inequalities of the last two named authors for variations of Riesz modulated energies along a transport, we prove uniform-in-time mean-field convergence in the gradient case with a rate which is sharp for the modulated energy pseudo-distance. For gradient dynamics, this completes in the periodic case the range $d-2\leq s<d$ not addressed by previous work of the second two authors. We also combine our relaxation estimates with the relative entropy approach of Jabin and Wang for so-called $\dot{W}^{-1,\infty}$ kernels, giving a proof of uniform-in-time propagation of chaos alternative to Guillin et al.

math.AP↗