arXiv subjects

Tobias Ried

Publications and source records attributed to Tobias Ried.

13 recordsLinked to original sources

arXiv2026

Large scale regularity and correlation length for almost length-minimizing random curves in the plane

We consider a model of random curves in the plane related to the large-scale behavior of the Random Field Ising Model (RFIM) at temperature zero in two space dimensions. Our work is motivated by attempts to quantify the Imry-Ma phenomenon concerning the rounding of the phase transition by quenched disorder, and connects to recent advances regarding the decay of correlations in the RFIM. We study a continuum model of minimal surfaces in two space dimensions subject to an external, quenched random field, and restrict ourselves to isotropic surface integrands. The random fields we consider behave like white noise on large scales with an ultra-violet regularization reminiscent of the lattice structure of the RFIM. We give a finer description of the minimizer below the length scale $ L_* $ starting from which the influence of boundary conditions is suppressed with a given probability, which has recently been shown to satisfy $ \log L_* \sim \varepsilon^{-\frac{4}{3}} $ in the amplitude $\varepsilon>0$ of the noise. More precisely, we prove flatness of the phase boundaries on scales $ L $ up to $ \log L \lesssim \varepsilon^{-\frac{4}{13}} $.

math-ph↗

arXiv2026

Partial regularity of optimal transport with Coulomb cost

We prove that for two-marginal optimal transport with Coulomb cost on $\mathbb{R}^d$, the optimal map is a $C^{1,α}$ diffeomorphism outside a closed set of Lebesgue measure zero provided the marginals are $α$-Hölder continuous, bounded, and strictly positive. Excluding a set of measure zero is necessary as optimal maps for the Coulomb cost have long been known to exhibit jump singularities across codimension $1$ surfaces (even for smooth marginals on convex domains).

math.AP↗

arXiv2024

On a variational problem related to the Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities

We explicitly solve a variational problem related to upper bounds on the optimal constants in the Cwikel--Lieb--Rozenblum (CLR) and Lieb--Thirring (LT) inequalities, which has recently been derived in [Invent. Math. 231 (2023), no.1, 111-167. https://doi.org/10.1007/s00222-022-01144-7 ] and [J. Eur. Math. Soc. (JEMS) 23 (2021), no.8, 2583-2600. https://doi.org/10.4171/jems/1062 ]. We achieve this through a variational characterization of the $L^1$ norm of the Fourier transform of a function and duality, from which we obtain a reformulation in terms of a variant of the Hadamard three lines lemma. By studying Hardy-like spaces of holomorphic functions in a strip in the complex plane, we are able to provide an analytic formula for the minimizers, and use it to get the best possible upper bounds for the optimal constants in the CLR and LT inequalities achievable by the method of [Invent. Math. 231 (2023), no.1, 111-167. https://doi.org/10.1007/s00222-022-01144-7 ] and [J. Eur. Math. Soc. (JEMS) 23 (2021), no.8, 2583-2600. https://doi.org/10.4171/jems/1062 ].

math-ph↗

arXiv2024

$p$-Wasserstein barycenters

We study barycenters of $N$ probability measures on $\mathbb{R}^d$ with respect to the $p$-Wasserstein metric ($1<p<\infty$). We prove that -- $p$-Wasserstein barycenters of absolutely continuous measures are unique, and again absolutely continuous -- $p$-Wasserstein barycenters admit a multi-marginal formulation -- the optimal multi-marginal plan is unique and of Monge form if the marginals are absolutely continuous, and its support has an explicit parametrization as a graph over any marginal space. This extends the Agueh--Carlier theory of Wasserstein barycenters [SIAM J. Math. Anal. 43 (2011), no.2, 904--924] to exponents $p\neq 2$. A key ingredient is a quantitative injectivity estimate for the (highly non-injective) map from $N$-point configurations to their $p$-barycenter on the support of an optimal multi-marginal plan. We also discuss the statistical meaning of $p$-Wasserstein barycenters in one dimension.

math.AP↗

arXiv2024

Domain branching in micromagnetism: scaling law for the global and local energies

We study the occurrence of domain branching in a class of $(d+1)$-dimensional sharp interface models featuring the competition between an interfacial energy and a non-local field energy. Our motivation comes from branching in uniaxial ferromagnets corresponding to $d=2$, but our result also covers twinning in shape-memory alloys near an austenite-twinned-martensite interface (corresponding to $d=1$, thereby recovering a result of Conti [Comm. Pure Appl. Math. 53 (2000), 1448-1474. https://doi.org/cgkvm4 ]). We prove that the energy density of a minimising configuration in a large cuboid domain $Q_{L,T}=[-L,L]^d\times [0,T]$ scales like $T^{-\frac{2}{3}}$ (irrespective of the dimension $d$) if $L\gg T^{\frac{2}{3}}$. While this already provides a lot of insight into the nature of minimisers, it does not characterise their behaviour close to the top and bottom boundaries of the sample, i.e. in the region where the branching is concentrated. More significantly, we show that minimisers exhibit a self-similar behaviour near the top and bottom boundaries in a statistical sense through local energy bounds: for any minimiser in $Q_{L,T}$, the energy density in a small cuboid $Q_{\ell,t}$ centred at the top or bottom boundaries of the sample, with side lengths $\ell \gg t^{\frac{2}{3}}$, satisfies the same scaling law, that is, it is of order $t^{-\frac{2}{3}}$.

math.AP↗

arXiv2024

$h$-Wasserstein barycenters

We generalize the notion and theory of Wasserstein barycenters introduced by Agueh and Carlier (2011) from the quadratic cost to general smooth strictly convex costs $h$ with non-degenerate Hessian. We show the equivalence between a coupled two-marginal and a multi-marginal formulation and establish that the multi-marginal optimal plan is unique and of Monge form. To establish the latter result we introduce a new approach which is not based on explicitly solving the optimality system, but instead deriving a quantitative injectivity estimate for the (highly non-injective) map from $N$-point configurations to their $h$-barycenter on the support of an optimal multi-marginal plan.

math.AP↗

arXiv2023

Variational methods for a singular SPDE yielding the universality of the magnetization ripple

The magnetization ripple is a microstructure formed in thin ferromagnetic films. It can be described by minimizers of a nonconvex energy functional leading to a nonlocal and nonlinear elliptic SPDE in two dimensions driven by white noise, which is singular. We address the universal character of the magnetization ripple using variational methods based on $Γ$-convergence. Due to the infinite energy of the system, the (random) energy functional has to be renormalized. Using the topology of $Γ$-convergence, we give a sense to the law of the renormalized functional that is independent of the way white noise is approximated. More precisely, this universality holds in the class of (not necessarily Gaussian) approximations to white noise satisfying the spectral gap inequality, which allows us to obtain sharp stochastic estimates. As a corollary, we obtain the existence of minimizers with optimal regularity.

math.PR↗

arXiv2021

Variational approach to regularity of optimal transport maps: general cost functions

We extend the variational approach to regularity for optimal transport maps initiated by Goldman and the first author to the case of general cost functions. Our main result is an $ε$-regularity result for optimal transport maps between Hölder continuous densities slightly more quantitative than the result by De Philippis-Figalli. One of the new contributions is the use of almost-minimality: if the cost is quantitatively close to the Euclidean cost function, a minimizer for the optimal transport problem with general cost is an almost-minimizer for the one with quadratic cost. This further highlights the connection between our variational approach and De Giorgi's strategy for $ε$-regularity of minimal surfaces.

math.AP↗

arXiv2018

Cwikel's bound reloaded

There are a couple of proofs by now for the famous Cwikel--Lieb--Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schrödinger operator, proven in the 1970s. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel's proof is said to give a constant which is at least about 2 orders of magnitude off the truth. This situation did not change much during the last 40+ years. It turns out that this common belief, i.e, Cwikel's approach yields bad constants, is not set in stone: We give a drastic simplification of Cwikel's original approach which leads to an astonishingly good bound for the constant in the CLR inequality. Our proof is also quite flexible and leads to rather precise bounds for a large class of Schrödinger-type operators with generalized kinetic energies. Moreover, it highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis.

math-ph↗

arXiv2017

Entropy decay for the Kac evolution

We consider solutions to the Kac master equation for initial conditions where $N$ particles are in a thermal equilibrium and $M\le N$ particles are out of equilibrium. We show that such solutions have exponential decay in entropy relative to the thermal state. More precisely, the decay is exponential in time with an explicit rate that is essentially independent on the particle number. This is in marked contrast to previous results which show that the entropy production for arbitrary initial conditions is inversely proportional to the particle number. The proof relies on Nelson's hypercontractive estimate and the geometric form of the Brascamp-Lieb inequalities due to Franck Barthe. Similar results hold for the Kac-Boltzmann equation with uniform scattering cross sections.

math-ph↗

arXiv2017

Solitary waves in nonlocal NLS with dispersion averaged saturated nonlinearities

A nonlinear Schrödinger equation (NLS) with dispersion averaged nonlinearity of saturated type is considered. Such a nonlocal NLS is of integro-differential type and it arises naturally in modeling fiber-optics communication systems with periodically varying dispersion profile (dispersion management). The associated constrained variational principle is shown to posses a ground state solution by constructing a convergent minimizing sequence through the application of a method similar to the classical concentration compactness principle of Lions. One of the obstacles in applying this variational approach is that a saturated nonlocal nonlinearity does not satisfy uniformly the so-called strict sub-additivity condition. This is overcome by applying a special version of Ekeland's variational principle.

math.AP↗

arXiv2015

Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction

We study weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules with a logarithmic singularity of the collision kernel for grazing collisions. Even though in this situation the Boltzmann operator enjoys only a very weak coercivity estimate, it still leads to strong smoothing of weak solutions in accordance to the smoothing expected by an analogy with a logarithmic heat equation.

math.AP↗

arXiv2015

Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules

It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties as the heat equation with a fractional Laplacian. In particular, the weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation with initial datum in $L^1_2(\mathbb{R}^d)\cap L\log L(\mathbb{R}^d)$, i.e., finite mass, energy and entropy, should immediately become Gevrey regular for strictly positive times. We prove this conjecture for Maxwellian molecules.

math.AP↗