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Felix Otto

Publications and source records attributed to Felix Otto.

At least 19 recordsLinked to original sources
arXiv2026

A geometrically linear approximation of min-max optimal matching

In this work we establish a connection between the min-max optimal matching in the critical dimension $d=2$ and a simpler, geometrically linear, action of random curves in a Brownian potential. This is done by following Leighton and Shor and working with the dual formulation, which we approximate by a problem of isoperimetric-type with a random volume term of white-noise character. This allows us to extract the leading-order term in the asymptotics of the expected cost, sharpening previous results.

math.PR↗
arXiv2026

Asymptotics of a planar isoperimetric problem with a white-noise volume term

In this work we study the maximal ratio $I$ between the white noise integrated over a set and the perimeter of the set, which can be seen as a random isoperimetric problem, and appears in the random-field Ising model (Ding-Wirth) and min-max optimal matching (Leighton-Shor). In the planar case considered here, such a ratio is scale-invariant and therefore the problem is critical. As such it requires an ultraviolet cutoff, which we impose by restricting to polygonal sets with side-length at least 1 contained in the ball $B_L$. Our main result establishes the leading-order asymptotics $\mathbb{E}I\approx i\ln^{3/4}L$ for some $i\in(0,\infty)$ and superconcentration at scale $O(\ln^{-1/4}L)$. This is done by relating to a simpler $(1+1)$-dimensional action, studied by the last two authors and C. Wagner. Such a connection is found by a classical geometric linearization of the perimeter to the Dirichlet energy, justified by Ried-Wagner large-scale regularity theory, and allows a coarse-graining and scale-by-scale iteration argument.

math.PR↗
arXiv2026

Tuning of Photoexcited Electron Dynamics at Monolayer h-BN/Metal Interfaces by Corrugation

Atomic-scale corrugation in two-dimensional materials can modify interfacial electronic coupling, yet its influence on ultrafast carrier relaxation remains poorly established. Here, we compare image potential states (IPS) at monolayer h-BN/Ir(111) and h-BN/Pt(111) interfaces using structural characterization and time-resolved two-photon photoemission spectroscopy. Consistent with literature, h-BN is strongly corrugated on Ir(111) but comparatively flat on Pt(111). The first (n = 1) and second (n = 2) IPS appears at similar energies on both substrates, whereas their relaxation dynamics differ markedly. On h-BN/Ir(111), the IPS decay is response-limited (<20 fs), while h-BN/Pt(111) exhibits lifetimes of 56 fs (n = 1) and 75 fs (n = 2). The lifetime contrast is most consistently explained by corrugation-enhanced overlap of the IPS wavefunction with the metal substrate, which accelerates electron decay. These results indicate that atomic-scale corrugation is an effective physical parameter for tuning ultrafast electron dynamics at two-dimensional material and metal interfaces.

cond-mat.mtrl-sci↗
arXiv2026

Sharp upper bound for a branched transport problem coming from Ginzburg-Landau models

We consider a branched transport type problem with weakly imposed boundary conditions, which can be seen as a blown-up version of a reduced model for type-I superconductors in the regime of vanishing external magnetic field. We prove that if the irrigated measure is (locally) Ahlfors regular then it is of dimension at most $8/5$ in agreement with the conjecture by Conti, the third author and Serfaty.

math.AP↗
arXiv2026

Quantitative homogenization of the maximal action of curves in a Brownian potential

Motivated by an optimal-matching problem (Leighton-Shor) and the random-field Ising model (Aizenman-Wehr, Ding-Wirth), we consider a variational problem for graphs in $1+1$ dimension maximizing an action that is the difference of a field term given by integrating white noise over the subgraph on the one hand, and the Dirichlet integral of the (continuum) height function $h=h(x)$ on the other hand. This problem is scale-invariant in law, and requires a small-scale cut-off which we implement by restricting to $h$ that are piecewise linear on intervals of size $1$ and vanish at $x=0,L$. We show that with overwhelming probability, the maximal action $A$ satisfies $A=a\ln L+O(1)$ for a deterministic constant $a\in(0,\infty)$. This can be considered as a homogenization result that is quantitative in an optimal way. The present result sharpens a recent qualitative homogenization result by the authors with C. Wagner; it does so by finding bounds for the action that are locally uniform in the boundary conditions. Like the earlier result, the present one relies on pointwise bounds on the optimizer, as provided by Dembin-Elboim-Hadas-Peled in a more general setting. In the earlier work, the small-scale cut-off also involved an explicit discretization of the field term, yielding a Brownian potential that is i.i.d. in $x\in\mathbb{Z}$; this had the benefit of allowing for a comparison argument, but is inconvenient for the coarse-graining used here.

math.PR↗
arXiv2026

Intermittency of geometric Brownian motion on $ \textbf{SL}(n) $

This short note is motivated by a recently discovered connection between a drift-diffusion process in $n$-dimensional Euclidean space with a divergence-free drift sampled from a stationary and isotropic Gaussian ensemble of critical scaling on the one hand, and a geometric Brownian motion on $\textbf{SL}(n)$ on the other hand. This can be seen as a tensorial form of a stochastic exponential; it thus is naturally intermittent, which transfers to the pair distance of the drift-diffusion process. In this note, we quantify the intermittency of the geometric Brownian motion $\{F_\tau\}_{\tau\ge0}$ on $\textbf{SL}(n)$ also in dimensions $n>2$. We do so in two (related) ways: 1) by identifying the exponential growth rate for the $2p$-th stochastic moment $\mathbb{E}|F_\tau|^{2p}$ with its anomalous dependence on $p$ (and $n$), and 2) by quantifying a non-tightness of $|F_\tau|^2/\mathbb{E}|F_\tau|^2$ as $\tau\uparrow\infty$. It is the second property that transmits to the drift-diffusion process. The arguments rely on stochastic analysis: We write $\{F_\tau\}_{\tau\geq 0}$ as the solution of $dF=F_\tau\circ dB$ with $\{B_\tau\}_{\tau\geq 0}$ a Brownian motion on the Lie algebra $\mathfrak{sl}(n)$. The arguments leverage isotropy: The diffusion projects onto the spectrum of the Gram matrix $G=F^*F$, as captured by ${\rm tr}G^p$.

math.PR↗
arXiv2026

Trimming of extreme votes and favoritism: Evidence from the field

Despite a large body of theoretical literature on voting mechanisms, there is no documented evidence from real-world panel evaluations about the effect of trimming the extreme votes on sincere voting. We provide the first such evidence by comparing subjective evaluations of experts from different countries in competitive settings with and without a trimming mechanism. In these evaluations, some of the evaluated subjects are experts' compatriots. Using data on 29,383 subjective evaluations, we find that experts assign significantly higher scores to their compatriots in panels without trimming. However, in panels with trimming, this favoritism is generally insignificant.

econ.GN↗
arXiv2025

A Critical Drift-Diffusion Equation: Intermittent Behavior via Geometric Brownian Motion on $ \textbf{SL}(n)$

This paper concerns the so-called diffusion in the curl of the 2d Gaussian free field, and its generalization to higher dimensions $n \geq 2$, building on the scale-by-scale homogenization approach developed recently by Chatzigeorgiou, Morfe, Otto, and Wang [13]. It begins by reformulating the approximation scheme of that work in terms of SDEs in the length scale $L$. This exposes an unexpected connection with a certain geometric Brownian motion on the special linear group $\textbf{SL}(n)$. The analysis of this process sheds light on the original problem, particularly as it pertains to intermittent behavior exhibited by the (averaged) Lagrangian coordinate.

math.PR↗
arXiv2025

Varying reference-point salience

The salience of reference points may theoretically influence the loss aversion mechanism in effort provision. However, we still lack a direct test from real competitive settings that uses exogenous variation to measure the effect of salience. We exploit a natural experiment where highly professional and incentivized individuals perform in real competitive settings with exogenous variation of reference-point salience. While a relevant reference point is salient in some cases, it is obscured in others, which, under reasonable assumptions, may affect individuals' expectations. This enables us to examine the effect of reference-point salience on the loss aversion mechanism in effort provision. Our regression discontinuity analyses reveal that individuals with positive expectations outperform those with negative expectations, but only when the reference point is salient.

econ.GN↗
arXiv2025

From Combinatorics to Partial Differential Equations

The optimal matching of point clouds in $\mathbb{R}^d$ is a combinatorial problem; applications in statistics motivate to consider random point clouds, like the Poisson point process. There is a crucial dependance on dimension $d$, with $d=2$ being the critical dimension. This is revealed by adopting an analytical perspective, connecting e.\,g.~to Optimal Transportation. These short notes provide an introduction to the subject. The material presented here is based on a series of lectures held at the International Max Planck Research School during the summer semester 2022. Recordings of the lectures are available at https://www.mis.mpg.de/events/event/imprs-ringvorlesung-summer-semester-2022.

math.PR↗
arXiv2025

On minimizing curves in a Brownian potential

We study a $(1+1)$-dimensional semi-discrete random variational problem that can be interpreted as the geometrically linearized version of the critical $2$-dimensional random field Ising model. The scaling of the correlation length of the latter was recently characterized in [12] and [13, Section 5]; our analysis is reminiscent of the multi-scale approach of the latter work and of [20]. We show that at every dyadic scale from the system size down to the lattice spacing the minimizer contains at most order-one Dirichlet energy per unit length. We also establish a quenched homogenization result in the sense that the leading order of the minimal energy becomes deterministic as the ratio system size / lattice spacing diverges. To this purpose we adapt arguments from [9] on the $(d+1)$-dimensional version our the model, with a Brownian replacing the white noise potential, to obtain the initial large-scale bounds. Based on our estimate of the $(p=3)$-Dirichlet energy, we give an informal justification of the geometric linearization. Our bounds, which are oblivious to the microscopic cut-off scale provided by the lattice spacing, yield tightness of the law of minimizers in the space of continuous functions as the lattice spacing is sent to zero.

math.PR↗
arXiv2025

Multi-index Based Solution Theory to the $\Phi^4$ Equation in the Full Subcritical Regime

We obtain (small-parameter) well-posedness for the (space-time periodic) $\Phi^4$ equation in the full subcritical regime in the context of regularity structures based on multi-indices. As opposed to Hairer's more extrinsic tree-based setting, due to the intrinsic description encoded by multi-indices, it is not possible to obtain a solution theory via the standard fixed-point argument. Instead, we develop a more intrinsic approach for existence using a variant of the continuity method from classical PDE theory based on a priori estimates for a new `robust' formulation of the equation. This formulation also allows us to obtain uniqueness of solutions and continuity of the solution map in the model norm even at the limit of vanishing regularisation scale. Since our proof relies on the structure of the nonlinearity in only a mild way, we expect the same ideas to be sufficient to treat a more general class of equations.

math.AP↗
arXiv2024

A Critical Drift-Diffusion Equation: Connections to the Diffusion on $\textbf{SL}(2)$

In this note, we connect two seemingly unrelated objects: On the one hand is a two-dimensional drift-diffusion process $X$ with divergence-free and time-independent drift $b$. The drift is given by a stationary Gaussian ensemble, and we focus on the critical case where a small-scale cut-off is necessary for well-posedness and the large-scale cancellations lead to a borderline super-diffusive behavior. On the other hand is the natural diffusion $F$ on the Lie group $\textbf{SL}(2)$ of matrices of determinant one. As a consequence of this connection, the strongly non-Gaussian character of $F$ transmits to how $X$ depends on its starting point.

math.PR↗
arXiv2024

A critical drift-diffusion equation: intermittent behavior

We consider a drift-diffusion process with a time-independent and divergence-free random drift that is of white-noise character. We are interested in the critical case of two space dimensions, where one has to impose a small-scale cut-off for well-posedness, and is interested in the marginally super-diffusive behavior on large scales. In the presence of an (artificial) large-scale cut-off at scale L, as a consequence of standard stochastic homogenization theory, there exist harmonic coordinates with a stationary gradient $F_L$; the merit of these coordinates being that under their lens, the drift-diffusion process turns into a martingale. It has recently been established that the second moments diverge as $\mathbb{E}|F_L|^2\sim\sqrt{\ln L}$ for $L\uparrow\infty$. We quantitatively show that in this limit, and in the regime of small P\'eclet number, $|F_L|^2/\mathbb{E}|F_L|^2$ is not equi-integrable, and that $\mathbb{E}|{\rm det}F_L|/\mathbb{E}|F_L|^2 $ is small. Hence the Jacobian matrix of the harmonic coordinates is very peaked and non-conformal. We establish this asymptotic behavior by characterizing a proxy $\tilde F_L$ introduced in previous work as the solution of an It\^{o} SDE w. r. t. the variable $\ln L$, and which implements the concept of a scale-by-scale homogenization based on a variance decomposition and admits an efficient calculus. For this proxy, we establish $\mathbb{E}|\tilde F_L|^4\gg(\mathbb{E}|\tilde F_L|^2)^2$ and $\mathbb{E}({\rm det}\tilde F_L-1)^2\ll 1$. In view of the former property, we assimilate this phenomenon to intermittency. In fact, $\tilde F_L$ behaves like a tensorial stochastic exponential, and as a field can be assimilated to multiplicative Gaussian chaos.

math.PR↗
arXiv2024

The Gaussian free-field as a stream function: continuum version of the scale-by-scale homogenization result

This note is about a drift-diffusion process $X$ with a time-independent, divergence-free drift $b$, where $b$ is a smooth Gaussian field that decorrelates over large scales. In two space dimensions, this just fails to fall into the standard theory of stochastic homogenization, and leads to a borderline super-diffusive behavior. In a previous paper by Chatzigeorgiou, Morfe, Otto, and Wang (2022), precise asymptotics of the annealed second moments of $X$ were derived by characterizing the asymptotics of the effective diffusivity $\lambda_L$ in terms of an artificially introduced large-scale cut-off $L$. The latter was carried out by a scale-by-scale homogenization, and implemented by monitoring the corrector $\phi_L$ for geometrically increasing cut-off scales $L^+=ML$. In fact, proxies $(\tilde\phi_L,\tilde\sigma_L)$ for the corrector and flux corrector were introduced incrementally and the residuum $f_L$ estimated. In this short supplementary note, we reproduce the arguments of the above paper in the continuum setting of $M\downarrow 1$. This has the advantage that the definition of the proxies $(\tilde\phi_L,\tilde\sigma_L)$ becomes more transparent -- it is given by a simple It\^{o} SDE with $\ln L$ acting as a time variable. It also has the advantage that the residuum $f_L$, which is a martingale, can be efficiently and precisely estimated by It\^{o} calculus. This relies on the characterization of the quadratic variation of the (infinite-dimensional) Gaussian driver.

math.PR↗
arXiv2024

Lecture notes on Malliavin calculus in regularity structures

Malliavin calculus provides a characterization of the centered model in regularity structures that is stable under removing the small-scale cut-off. In conjunction with a spectral gap inequality, it yields the stochastic estimates of the model. This becomes transparent on the level of a notion of model that parameterizes the solution manifold, and thus is indexed by multi-indices rather than trees, and which allows for a more geometric than combinatorial perspective. In these lecture notes, this is carried out for a PDE with heat operator, a cubic nonlinearity, and driven by additive noise, reminiscent of the stochastic quantization of the Euclidean $\phi^4$ model. More precisely, we informally motivate our notion of the model $(\Pi,\Gamma)$ as charts and transition maps, respectively, of the nonlinear solution manifold. These geometric objects are algebrized in terms of formal power series, and their algebra automorphisms. We will assimilate the directional Malliavin derivative to a tangent vector of the solution manifold. This means that it can be treated as a modelled distribution, thereby connecting stochastic model estimates to pathwise solution theory, with its analytic tools of reconstruction and integration. We unroll an inductive calculus that in an automated way applies to the full subcritical regime.

math.PR↗
arXiv2023

Lifshitz Transition and Band Structure Evolution in Alkali Metal Intercalated 1Tprime-MoTe2

In van der Waals materials, coupling between adjacent layers is weak, and consequently interlayer interactions are weakly screened. This opens the possibility to profoundly modify the electronic structure, e.g., by applying electric fields or with adsorbates. Here, we show for the case of the topologically trivial semimetal 1Tprime-MoTe2 that potassium dosing at room temperature significantly transforms its band structure. With a combination of angle-resolved photoemission spectroscopy, scanning tunneling microscopy, x-ray photoemission spectroscopy, and density functional theory we show that i) for small concentrations of K, 1Tprime-MoTe2 undergoes a Lifshitz transition with the electronic structure shifting rigidly, and ii) for larger K concentrations 1Tprime-MoTe2 undergoes significant band structure transformation. Our results demonstrate that the origin of this electronic structure change stems from alkali metal intercalation.

cond-mat.mtrl-sci↗