Papers

We investigate long-time learning in ergodic, potential, monotone mean-field games (MFGs) via a self-fictitious-play (SFP) dynamics coupling an optimally controlled diffusion with a slowly evolving belief. At each time, the state follows the optimal feedback associated with the current belief, while the belief is updated using the player's own empirical occupation measure rather than the population distribution. For ergodic monotone potential MFGs on the torus, we prove that the SFP dynamics is contractive and admits a unique invariant law. Moreover, we show that this invariant law is quantitatively close to the MFG Nash equilibrium, with an error of order equal to the square root of the belief-update rate. The proof combines uniform-in-time regularity estimates for the ergodic Hamilton-Jacobi-Bellman equation with an energy argument based on the Lasry-Lions divergence. The linear-quadratic example shows that this rate is sharp, and the numerical experiments illustrate the predicted scaling.

The paper contains An elementary proof of Marton and Shields' obstruction to finitary coding.

We uncover the resolvent origin of the spectral duality governing reset-neutral distributions in Markov chains with geometric resetting. Starting from the abstract conditions of Paper~III, we show that the spectral duality $B_\nu(z)=\kappa(z)\,A_\nu(\sigma(z))$ is equivalent to a single symmetry of the resolvent $R(\gamma)=(I-(1-\gamma)P)^{-1}$: the intertwining relation $[\Delta^2\mathcal{R},R(\gamma)]=0$, where $\mathcal{R}$ is the reflection operator of an involution $\sigma$ and $\Delta=\operatorname{diag}(\sqrt{\kappa(z)})$; equivalently, $\widetilde{\mathcal{T}}=K^{-1/2}\Delta^2\mathcal{R}$ is an involution. This symmetry determines the universal critical value $C^*=1/(1+\sqrt{K})$, with $K=\kappa(z)\kappa(\sigma(z))$, which depends only on the scalar $K$ — not on the resetting rate $\gamma$, the reset distribution, or the particular chain. We characterize the class of $(\sigma,\kappa)$-reversible chains, encompassing both the biased random walk and genuinely non-homogeneous dynamics sharing the same $C^*$; a Doob $h$-transform realizes the duality $K\mapsto1/K$, hence $C^*\mapsto1-C^*$, with fixed point $C^*=1/2$. The orientation field admits the explicit resolvent representation $\psi(\gamma)=R(\gamma)(b^{(0)}-C^*b)$: its gauge-normalized form $\Delta^{-1}\psi(\gamma)$ is antisymmetric under $\sigma$, it has an exact node at the fixed point of $\sigma$, and it governs the exact sign law $\operatorname{sgn}(C(\pi,\gamma)-C^*) =\operatorname{sgn}\langle\pi,\psi(\gamma)\rangle$. Numerical experiments confirm the theory to machine precision. These results establish the operator-theoretic foundation of the spectral duality of Paper~III and provide the bridge to the information-geometric framework of Paper~V.

I study the largest eigenvalue \(\Lambda_j\) of \(X^{(j)}(X^{(j)})^*\), where \(X^{(j)}\) is the \(M_j\times j\) northwest rectangle of one infinite complex Gaussian array. For two upper-tail events and, separately, for two lower-tail events at the largest-eigenvalue soft edge, I prove finite-\(N\) one-sided product-scale upper bounds for their joint probabilities. The logarithmic walls have polynomially small marginals, so an additive \(o(1)\) covariance estimate may be much larger than the vanishing product that must be controlled. Uniformly for levels in a short macroscopic window, bounded deterministic shifts, and separations at least \(N^{2/3+\epsilon}\), each joint probability is at most \((1+o(1))\) times the product of its marginals. The classical \(N^{2/3}\) correlation window corresponds to order-one extended-Airy time; the theorem works at supercritical separation. Exact Laguerre operators along admissible row–column paths provide the finite-dimensional input. The upper-tail proof uses occupancy counts and cross-block trace estimates, whereas the lower-tail proof uses gap determinants, diagonal resolvents, and a Schur complement. I transfer the result between the half-integer Laguerre and physical rectangular normalizations and deduce intrinsic first- and second-moment estimates for separated sparse grids under the rarity, count-growth, spacing, and macroscopic-window budget.

We prove a quantitative central limit theorem for linear functionals of regularized empirical-risk minimizers in the proportional-dimensional regime \(p=O(n)\). The data columns are independent, not necessarily identically distributed, and satisfy a uniform columnwise Poincaré inequality. Under uniform curvature and smoothness assumptions, and for a quadratic regularizer, we show that every nondegenerate statistic \(\sqrt n\,u^\top\hat\theta\), centered by its expectation and normalized by its standard deviation, converges to a standard normal random variable in Wasserstein distance, with rate \(O((\log n)^7n^{-1/4})\). The proof is based on moment and stability bounds for the minimizer, a second-order leave-one-out expansion, and a perturbative normal-approximation argument for functions of independent variables. We also prove the variance upper bound \(\Var(u^\top\hat\theta)\le C\norm{u}_2^2/n\), identifying the \(\sqrt n\) fluctuation scale.

We construct an optimal transport map between uniform distributions on two antipodal annuli of a sphere.

In this paper we present a discrete version of Bertrand's paradox, where each sample space is a finite set of chords and a chord is chosen uniformly at random.

We introduce the category $\mathbf{SSS}$ of sample-smooth spaces over a mixed site. The test objects are the products $\Omega_n := \mathbb{R}^n \times \Omega$ of a Cartesian space with the universal Hilbert cube $\Omega$ carrying all universally measurable sets, and a space is a set with a family of admissible plots $\Omega_n \to \mathcal{X}$ closed under precomposition. Smoothness and measurability are then not two structures glued along an axiom, but one structure over one site. The site has finite non-empty products, because $\Omega$ absorbs its own square; its Karoubi envelope contains every $\mathbb{R}^n$; and it has mixed morphisms $\omega \mapsto (W(\omega),\Phi(\omega))$, which turn measurability of a smooth family from an axiom into a consequence. $\mathbf{SSS}$ is a concrete quasitopos: complete, cocomplete, cartesian closed and locally cartesian closed, with a classifier for embeddings. Morphisms of Cartesian spaces are exactly the $C^\infty$ maps and manifolds embed full and faithfully, both without Boman's theorem. Every object has tangent and cotangent spaces, every morphism a differential. The modalities sit in an adjoint string $\Pi \dashv \flat \dashv \natural \dashv \sharp \dashv \Lambda$, making $\mathbf{SSS}$ cohesive over quasi-universal spaces. The point is the probability monad. Defining the plots of $\mathsf{P}(\mathcal{X})$ as push-forwards of $\mathcal{X}$-plots at every test object, $\mathsf{P}$ is an unconditional strong commutative affine monad on all of $\mathbf{SSS}$ – functor, unit, product of kernels, multiplication and the monad laws are each one line of seed splitting – and its Kleisli category, of differentiable simulators, is a Markov category. The reparametrisation trick holds by construction: every Kleisli morphism is plot-wise a sampler, stably under composition. A reflection theorem locates the whole gain in a single plot family.

Training datasets for upcoming LLMs would include a significant amount of AI text/image data generated from current LLMs. In such a scenario, it is important to understand how this affects batch decompositions and thereby, the performance of the resultant new LLM. In this paper, we consider AI generated data as anomalies “linked" to main data points and study decomposition and undersampling properties of the overall random dataset. We use redundancy graphs and iteration techniques to obtain bounds for the minimum size of a strongly dissimilar (SD) decomposition and demonstrate a phase transition phenomena, wherein the minimum size is essentially determined by the main data points when the number of anomalies is small and is “taken" over by the anomalies above a certain threshold. We also establish a size criticality result for the strong similarity of a randomly undersampled dataset and illustrate our results with examples involving categorical datasets, whose overall space size is much larger than the size of the dataset.

We introduce a stochastic process for modeling the evolution in time of compositional measurements (i.e., a vector of non-negative values that add up to a total of 1). This model is a diffusion, as it is defined as the solution for a stochastic differential equation in the Ito sense, and it has a Dirichlet distribution as its steady distribution. We have named this process Dirichlet Diffusion (DD). As the process is confined to a manifold and the coefficients of the equation are not globally Lipschitz, the usual theorems do not apply directly and establishing the existence and properties of solutions requires a somewhat delicate analysis. We establish the existence of strong solutions under the assumption that all the parameters of the Dirichlet distribution are greater than 2; for the general case we were only able to establish the existence of weak solutions, but also that these solutions remain confined to the closed simplex without need for reflecting boundaries. A useful feature of DD that it inherits from the Dirichlet distribution is the property of aggregation: If components are combined to create a coarser composition, the resulting process is also a DD. This makes it useful, for example, for jointly modeling the evolution of a microbiome grouping the microbe species at different taxonomic levels.

The Zaporozhets–Tarasov conjecture is considered, which states that the mean distance between two random points inside a convex body does not exceed the mean distance between two random points on its boundary. For centrally symmetric bodies the question is settled completely: the conjecture is proved for planar bodies and disproved in every dimension at least three, for all moments of the distance; an explicit family of counterexamples is constructed together with a dimension-lifting construction. It is also shown that for sufficiently high moments an analogous inequality holds for any planar convex body. For circumscribed bodies exact relations between the mean distances are obtained — analogues of Kingman's formula derived via the projective distributions introduced here; moreover, the conjecture is proved for triangles in a stronger per-projection form.

We prove that the mean distance between two independent uniformly distributed points in an arbitrary planar convex body is strictly smaller than the mean distance between two independent uniformly distributed points on its boundary. Equality is impossible, although the difference between these two mean distances tends to zero along a sequence of thin rectangles. The proof reduces the problem to a one-dimensional comparison of two Gini mean differences. The main new ingredient is a moment lemma for a random pair $(\varepsilon,R)\in\{-1,1\}\times[0,1]$. The only finite algebraic part of the proof is given by exact rational certificates in the Bernstein basis; the verification script and all 6492 coefficients are available in the supplementary materials.

We study finite-alphabet spin models on trees evolving under independent symmetric spin-flip dynamics. On the lattice the time-evolved plus measure of the low temperature Ising model in zero external field was shown to be non-quasilocal for all sufficiently large times in \cite{EnFeHoRe02}. On the tree however the time-evolved plus phase of the Ising model behaves differently, as the quasilocal Gibbs property is first lost, but then recovered at a later time \cite{EnErIaKu12}. We first extend this result by showing that large-time reentry into the quasilocal Gibbs regime on the tree happens more generally for all finite alphabet models, when the dynamics is started in $f$-stable Gibbs states. These are defined in terms of a time-independent sharp condition on the homogeneous recursion $f$ describing the model, which can be checked explicitly. As our second and opposite result, we develop a method for proving large-time persistence of non-quasilocality of time-evolved measures. We show that even all configurations can become bad at large times, when the initial Gibbs measure corresponds to an $f$-saddle and give an explicit illustration for the Potts model. In our proofs we study the influence of spatially inhomogeneous perturbations to the solutions of a time-dependent fixed point problem.

We study a general class of gradient interface models with Hamiltonian $H=\beta\sum V(\nabla\phi)$, $\beta>0$, assuming essentially that the potential $V$ is even, $V'(s)\ge \alpha s$ on $[0,\infty)$ for some $\alpha>0$, and $-M\leq V''\le C$. We establish a Helffer-Sjöstrand representation for these models, and use it to prove that their scaling limits are Gaussian Free Fields (GFFs), and that their covariances decay at the same rate as the GFF. This extends results for strictly convex potentials to a large class of non-convex potentials and to arbitrary temperatures. Additionally, we prove Brascamp-Lieb and dimension-free Poincaré inequalities for the models. We obtain these results by representing the interface as a mixture of gradient interface models with strictly convex potentials, extending an idea by Biskup-Spohn who had considered mixtures of Gaussians at moderate inverse temperature $\beta=1$. The construction of such a representation is one of the key new contributions of this work.

We introduce and study a Fisher information metric \(g^F_\tau\) associated to the conjugate heat kernel of a Ricci flow \((M^n,g_t)\). This tensor measures, at a fixed scale, how the pointed heat-kernel measure changes when the base point is moved. We prove that \(g^F_\tau\) is monotone in scale and satisfies \(g^F_\tau\le g_t\). We relate its trace to the pointed Nash entropy and prove a matrix square identity for the Fisher defect \(g_t-g^F_\tau.\) This identity gives a rigidity theorem for the equality case; on closed connected flows one has the strict inequalities \(0<g^F_\tau<g_t\) at every positive scale, while in the complete case equality forces a Euclidean splitting. We also develop several consequences of this point of view. These include a sharp reverse Poincaré inequality for the heat semigroup, a contraction formula for \(\varphi\)-divergences along conjugate heat flow, and a canonical construction of heat-kernel splitting maps from large eigenvalues of the Fisher endomorphism. As applications, we relate pointed Nash entropy close to \(0\) to small Fisher deficit at comparable scales, and we obtain a codimension-one Fisher-metric criterion for applying Bamler's \(\varepsilon\)-regularity theorem.

Let $P$ be a real polynomial of degree at most $m$ on $\mathbb{R}^d$. We construct finitely many correlated Gaussian moments that determine $P$ up to an orthogonal change of variables. The number of copies and the rational covariance matrices are specified in terms of $d,m$. On coefficient balls, the induced map on orthogonal orbits has a Hölder continuous inverse with an explicit exponent. For $m\geq3$, the normalised local stationary-phase expansion at the critical branch through the origin equals the Gaussian moment series. Its coefficients recover the same finite moment vector, without assumptions on the other critical points. If the leading homogeneous part has an isolated complex critical point, we compute the twisted de Rham ranks and construct formal lattices on each fixed coupling support.

The paper focuses on information theoretic converse bounds for the alignment of $m$ correlated graphs and for the detection of correlation among $m$ graphs. A simple idea for $m\geq 3$ is that if the alignment of $m-1$ of the graphs is revealed as extra information (by a genie for example) then it is still necessary to produce the alignment between the one remaining graph and the others, i.e. the last matching must be accomplished. For both Gaussian and Erdos-Renyi models, the last-matching problem is equivalent to one with two observed graphs, providing a path to extend converse bounds for $m=2$ to larger $m$. While the method is rather obvious for alignment, we show that the method can also be used to derive converse bounds for weak detection of correlation.

We consider the notion of shattering introduced by Ben Arous and Jagannath for spherical pure $p$-spin glasses with overlap $q$. For every $p\geq 3$ and $0<\beta\leq\beta_{\mathrm{sh}}(p)$, we rule out shattering whenever $q\leq2^{-1/2}$ or $q>\sqrt{(p-2)/(p-1)}$. The proof combines a deterministic $N+1$ bound for disjoint bands in the first range with a general-$p$ sign law showing that their total marked weight has subdominant free energy in the second. A spherical-code bound and Hölder's inequality give an additional $q$-dependent obstruction; in particular, they rule out every fixed overlap for $0<\beta\leq\sqrt{\log2}$. For $p=3$, the first two ranges already exhaust every fixed $q\in(0,1)$, so the landscape is not shattered at any $T\geq T_{\mathrm{sh}}$. For $p\geq4$, the cases not covered by our criteria are confined to $2^{-1/2}<q\leq\sqrt{(p-2)/(p-1)}$ and $\sqrt{\log2}<\beta\leq\beta_{\mathrm{sh}}(p)$. In particular, this paper partially resolves Conjecture 1 of the paper above and also suggests new methods to show non-shattering.

We give a deterministic proof of Bourgain's slicing theorem. The proof uses neither stochastic localization nor stochastic calculus, does not rely on Guan's covariance bound, and avoids $M$-ellipsoids, Pisier's regularization machinery, and complex interpolation. Instead, the argument is based on a deterministic one-parameter quadratic perturbation of the measure, a covariance-survival principle, and a geometric analysis of $L_p$-centroid bodies.

While infinite-dimensional inverse problems are traditionally analyzed assuming continuous data, practical applications rely on finitely many discrete measurements. Recent deterministic approaches establish that unknowns belonging to a $d$-dimensional model class can be stably recovered from finitely many measurements. However, for severely ill-posed problems, such as the Calderón problem and inverse scattering, the known constructions may require a number of measurements that is exponential in $d$. We show that random sampling reduces this count dramatically if one asks only for exact identifiability. By exploiting the analytic geometry of the forward maps, we prove that, whenever the infinite-data problem is injective on the model class, $2d+1$ random scalar measurements determine the unknown uniquely, almost surely. Applications are given to the Calderón problem, with both infinite- and finite-dimensional boundary sampling, and to inverse medium scattering from randomly sampled far-field values.

We study Ollivier–Ricci curvature on Cayley graphs of groups of polynomial growth. Our main result shows that non-negative Ollivier–Ricci curvature forces the group to be virtually abelian. As an application, we prove that connected vertex-transitive graphs of polynomial growth and non-negative Ollivier–Ricci curvature are quasi-isometric to $\mathbb{Z}^k$, for some $k\in\mathbb{N}$.

We examine the modulus of continuity, in the spatial variable, of the local time process of Feller Brownian motion (FBM) on the half-line $[0,\infty)$. Briefly, a FBM is a strong Markov process on $[0,\infty)$ that moves like standard Brownian motion on $[0,\infty)$ up until it first encounters the state $0$. The process returns to $(0,\infty)$, either continuously (like reflecting Brownian motion) or by jumping to a (random) positive state chosen according to a specified measure. The present work is a continuation and application of our earlier work with Michael Marcus on the moduli of continuity for the local times of a Markov process built by piecing together (“rebirthing") the paths of another Markov process with finite lifetime. We first establish a general result on the resolvent and local times for a rebirthed process with a special holding state (the state $0$ for FBM). We show how our earlier approach using the Eisenbaum Isomorphism Theorem on an assemblage of excursions works out in this context. This knowledge is then used as an approximation device to obtain our main result on the exact uniform moduli of continuity for the local time of FBM on a spatial interval of the form $(0,1]$. Extensions are made, under certain conditions, to the more delicate situation of the spatial interval $[0,1]$. We also consider briefly the case of more general diffusions on $[0,\infty)$.

We consider simple autoregressive processes of type AR(1), whose stationary distribution is supported on a subset of the unit interval and is an affine transformation of a Bernoulli convolution. A new structural representation of the stationary distribution as a product of a power function with a log-periodic function near the origin is given, which gives structural insight to the stationary distribution on the whole unit interval by using a characteristic functional equation. This enables to prove that the stationary distribution of the AR(1)-process belongs to the domain of geometric partial attraction of a max-semistable law. We further prove uniform convergence of the distribution function of normalized maxima of the AR(1)-process to a certain power of the max-semistable law in the spirit of a merge theorem and point out connections to deterministic and random dynamical systems.

We consider a sequence of discrete random variables whose generating functions are polynomials only having roots on the unit circle in the complex plane. For such sequences we prove, under mild assumptions, mod-$\phi$ convergence with explicit rate and limiting function. This allows us to recover, in a unified way, classical central limit theorems as well as finer asymptotic expansions. The general theory is applied to random permutations and random partitions.

In the first part we study excessive functions for a Markov process that are continuous semimartingales along the sample paths. The property then follows from the theory of local times of semimartingales. We next treat the case of Dirichlet spaces and show that, if u is a quasi-continuous version of a function u belonging to a regular Dirichlet space, then the image under u of the local energy measure of u is absolutely continuous with respect to Lebesgue measure. Finally, in a third part, we establish the occupation-time density property for certain Dirichlet processes.

A non-scale invariant BHP on any open set is obtained for a large class of discontinuous Hunt processes on metric measure spaces that are in weak duality with another Hunt process under suitable conditions in terms of estimates of exit distributions from open sets, and bounds on Green functions and the comparability of the jump density function. These conditions are easy to verify in concrete cases. Several examples are given to illustrate the main results and the new contributions of this paper. Our result in particular establishes the BHP on any open set for any Hunt process associated with a regular symmetric Dirichlet form on a metric measure space having both the strongly local term and the pure-jump term that admits a two-sided heat kernel estimates of the mixture of (sub-)Gaussian and stable-like form, as well as for a wide class of non-symmetric diffusion processes with jumps on $\mathbb R^d.$

We give an approximate sampler for ferromagnetic Ising models with no field on arbitrary graphs that runs in time $\widetilde O(m+n)+\widetilde O_\beta(n^2\log^2 (1 / \varepsilon))$, where $n$ and $m$ are the numbers of vertices and edges, respectively, and $\varepsilon$ is the approximation error. Our approach combines Benczúr–Karger cut sparsification with a new mixing time analysis of the Glauber dynamics for the random-cluster model. The mixing time analysis features a new monotone edge-count Poincaré inequality.

The Johnson–Lindenstrauss lemma asserts that every set of $n$ points in $d$-dimensional Euclidean space embeds into $O(\varepsilon^{-2}\log n)$-dimensional Euclidean space with distortion at most $1+\varepsilon$. Larsen and Nelson conjectured that the optimal target dimension throughout the full range of the parameters $n,d, \varepsilon$ is \[ \Theta\left(\min\left\{d,n-1,\frac{\log(2+\varepsilon^2n)}{\varepsilon^2}\right\}\right). \] We resolve this conjecture in the affirmative. In fact, we prove the stronger statement that the upper bound is attained by a linear map. The matching lower bound, due to Larsen–Nelson and Alon–Klartag, holds even for nonlinear embeddings.

Using the local time-space calculus of Peskir (2005) and the method developed in Mijatovic (2010), we derive a new integral representation for the backward probability associated with the first-passage time (FPT) of a diffusion process through a time-dependent barrier. We present a complete three-step numerical algorithm: first, the problem is reduced to a Volterra-type integral equation; second, its kernel is approximated by a Markov chain; finally, the resulting equation is solved using a quadrature rule. The method is implemented for several representative examples, and its convergence properties are established. An extension to double barrier problems is carried out

We prove a hitting-time analogue of cutoff for the star transposition shuffle on the symmetric group S_n. Let tau be the first time at which every non-top card has been selected. We show that the shuffle is asymptotically mixed at time tau: more precisely, the total variation distance between the law of Y_tau and the uniform distribution on S_n is at most exp(-(log n)^(1/2+o(1))). Our proof compares the star transposition shuffle with the random transposition shuffle using simultaneous diagonalization of the two transition kernels, and then adapts the hitting-time strategy of Jain and Sawhney. This introduces a technique that can be applied to card shuffles that are not necessarily conjugacy invariant.