Papers

A conjecture of Komlós states that the combinatorial discrepancy of any matrix $A\in\mathbb R^{m\times n}$ whose columns have Euclidean norm at most one is bounded by a universal constant. We prove that the combinatorial discrepancy of every such matrix is at most $O((\log n)^{1/4}(\log\log n)^{7/4})$. This is the first asymptotic improvement over the $O(\sqrt{\log n})$ bound established by Banaszczyk [Banaszczyk, Random Struct.\ Algorithms, 1998], and it refutes a conjecture of Hajela [Hajela, European J.\ Combin., 1988] that a lower bound of order $\Omega(\sqrt{\log n})$ should hold.

We study which probability measures on $[0,\infty)$ can occur as the distribution of the distance between two independent points sampled from a complete separable metric space equipped with a Borel probability measure. We prove that an atomless Borel probability measure can be realised in this way if and only if its support contains the origin. This settles the conjecture of Aldous, Blanc, and Curien for absolutely continuous laws and also covers atomless measures that are singular with respect to Lebesgue measure. Moreover, for every $L>1$, we show that the realising metric may be chosen $L$-bi-Lipschitz equivalent to an ultrametric. The proof constructs the space from compact components whose mutual distances produce prescribed parts of the measure, while distances within the components are assigned to smaller scales.

We prove that the Fourier coefficients of the canonical critical Gaussian multiplicative chaos on the circle vanish almost surely at infinity. More precisely, let $M_\phi^{\mathrm{crit}}$ be the canonical critical chaos associated with the centered circle field $\phi$ of covariance $\mathbb{E}[\phi(\theta)\phi(\theta')] = \log\frac{1}{\lvert e^{i\theta}-e^{i\theta'}\rvert}$. Then, almost surely, $\widehat{M_\phi^{\mathrm{crit}}}(n)\longrightarrow0$ as $\lvert n\rvert\to\infty$. This resolves the almost-sure critical Rajchman problem for the canonical circle field. Since critical chaos has Fourier dimension zero almost surely, no positive polynomial Fourier-decay rate can hold; the theorem therefore exhibits qualitative Fourier cancellation beyond the regime of positive Fourier dimension. The proof addresses two coupled difficulties: the heavy, nonuniform cell masses of critical chaos and the need to control exponentially many frequencies in each dyadic annulus. For an auxiliary periodized compact-range star-scale field, a derivative-rooted Bessel regression yields weighted small-cell summability and moving-tail control of exceptional large cells. After conditioning at a coarse scale below the Fourier scale, finite-range independence and conditional Bernstein concentration reduce uniform control of the terminal Fourier coefficients over each dyadic annulus to a spatial-variation estimate for a coarse predictable measure. A smooth positive-definite covariance correction and critical-chaos uniqueness then transfer the Rajchman property to the canonical critical chaos of the exact circle field.

We establish local well-posedness for a class of non-linear heat equations on the torus $\mathbb{T}^n$ whose highest order non-linearities have scaling-critical dimension $-1/k, \, k\in \mathbb{N}$, when the initial condition is a random field satisfying a spectral gap inequality involving a suitable Lebesgue norm. The corresponding subcriticality condition matches that of the Gaussian free field in dimension $d<2+2/k$. This extends previous subcritical well-posedness results beyond the Gaussian setting. Moreover, our proof is simpler and avoids diagrammatic arguments.

We study the characteristic polynomial of self-normalized random matrices, whose rows are independent and normalized to have unit $\mathrm{L}^2$ norm. The entries before normalization are assumed to have regularly varying tails with tail index $\alpha \in [0,2]$. We prove that, outside the unit disk, the characteristic polynomial converges to a random analytic function $F_\alpha$. We identify $F_\alpha$ as a multiplicative chaos described in terms of Poisson point processes. The family of limiting functions $(F_\alpha)_{\alpha\in[0,2]}$ interpolates between two universal regimes: Poisson multiplicative chaos at $\alpha=0$ and Gaussian multiplicative chaos at the boundary $\alpha=2$. Thus, self-normalization provides a matrix model where one observes the transition between Poissonian and Gaussian regimes for the limiting characteristic polynomial as the tail index varies. A similar transition is found for the fluctuations of the traces of self-normalized matrices. As an application of our results, we derive that the spectral radius of self-normalized matrices is asymptotically bounded above by one in probability for any symmetric entry distribution.

We consider a class of discrete multi-time determinantal point processes whose correlation kernels admit a double-contour integral form, containing Schur processes as simple examples. We show that the form of the correlation kernel is preserved under conditioning of the point process to a restricted domain, however, with an integrand which becomes more complicated and which is characterized by a Riemann-Hilbert problem. We apply our general result to domino tilings of reduced Aztec diamonds and to lozenge tilings of hexagons with holes. Our results show a striking analogy with the Its-Izergin-Korepin-Slavnov method for integrable kernels.

Let $d\geq 2$, $0<\alpha<1$, and let $\Gamma_k$ be the successive arrival times of a standard Poisson process on $(0,\infty)$. Given independent uniform directions $\varepsilon_k\in S^{d-1}$, independent of $(\Gamma_k)$, we consider the random countable stable zonotope $Z_\alpha=\bigoplus_{k=1}^{\infty}\Gamma_k^{-1/\alpha}[0,\varepsilon_k]$. For its set of extreme points $\operatorname{ext} Z_\alpha$, we prove that almost surely $\dim_H \operatorname{ext} Z_\alpha=(d-1)\alpha$, and that the critical Hausdorff measure $\mathcal H^{(d-1)\alpha}(\operatorname{ext} Z_\alpha)$ is almost surely finite. The lower bound follows from the tangential non-degeneracy of the stable increments of the parametrizing field and Frostman's energy criterion. For the upper bound we construct an adaptive covering: at each scale the Poisson jumps are split into large and small ones, the large jumps determine a finite hyperplane arrangement, and the sum of the small jumps controls the diameters of the images of its cells.

The Khintchine inequality provides a lower bound on the expected absolute value of a weighted sum of independent Rademacher random variables. In the classical setting, when the weight vector has unit norm, this lower bound is a constant, with equality attained only for a simple family of extremal configurations. A refined version due to De, Diakonikolas, and Servedio (2013) – referred to as the linear Khintchine inequality – strengthens this by establishing a lower bound that depends linearly on the distance of the weight vector from the extremal set. In this paper, we present a refined analysis of this dependence on the weight vector. Our results reveal a phase transition in the rate of improvement: when the dimension exceeds six, the lower bound undergoes an abrupt change as the weight vector deviates from the minimizer. Additionally, we improve the slope constant in linear Khintchine inequality. As a consequence, we establish an improved lower bound on the degree-1 Fourier weight for linear threshold functions $\mathbf{W}^{\leq 1}[\mathrm{LTF}] \geq 0.53317$, marking progress towards a conjecture of O'Donnell.

We investigate the extremal statistics of planar $\beta=2$ Coulomb gases with radial external fields. For the rightmost eigenvalue and the spectral radius, we establish sharp Berry–Esseen bounds for their convergence to the Gumbel distribution, with explicit rates \[ \frac{25\log\log n}{4e\log n} \quad\text{and}\quad \frac{2\log\log n}{e\log n}, \] respectively. In addition, we derive sharp asymptotic equivalences for the large and moderate deviations of both statistics across all relevant scales. Analogous results hold for the smallest modulus.

We study the continuum scaling limit of the Langevin dynamics for three-dimensional U(1) lattice Yang–Mills theory. The model is defined on the discrete 3D torus with a general class of plaquette actions that are suitably normalized, including Wilson, Manton, and Villain actions. Under the weak-coupling scaling and in the DeTurck gauge, we prove that the rescaled logarithmic field converges to the solution of the one-form stochastic heat equation in probability. In particular, the limiting dynamics are universal and do not depend on the higher-order details of the plaquette action.

We study long-run optimal intervention strategies for endemic infections in a stochastic susceptible-infected-susceptible (SIS) model. The proportion of infected individuals evolves as a diffusion process with random fluctuations, while a control variable $\zeta_t\in[0,\tilde{\zeta}_{max}]$ represents the intensity of public health interventions that reduce transmission for some intervention threshold $\tilde{\zeta}_{max}\in (0,1]$. The objective is to minimize the long-run average societal cost, balancing the burden of infection against the costs of interventions. Under a concave intervention cost structure the problem can be formulated as an ergodic stochastic control problem, whose structure implies (under certain additional conditions) that optimal interventions are of bang-bang type, switching between no intervention and the maximal admissible intervention at a single switching threshold in the infection level. We construct candidate value functions, rigorously verify optimality in this single-threshold case, and relate the results to extinction and persistence properties of the underlying SIS dynamics in the absence of control. In our framework, the analysis provides a rigorous justification for the threshold-based intervention rules commonly used in epidemic management.

Let $a,b$ be integers with $1\le b\le a$, and let \[ G_0=1,\qquad G_1=a+1,\qquad G_{n+2}=aG_{n+1}+bG_n. \] For real-valued functions which are additive in the greedy $G$-digits, we prove a necessary-and-sufficient criterion for the existence of a limiting distribution. The criterion consists of a first-order drift series and a quadratic digit-energy series, and it recovers the Zeckendorf theorem when $a=b=1$. The main difficulty is necessity: a one-step transfer matrix detects all non-maximal digit values, while the maximal digit becomes visible only in a two-step product. A weighted Euclidean norm symmetrizes the untwisted companion matrix and makes both contractions occur at the true Perron scale. Sufficiency follows from a two-dimensional Perron product lemma with square-summable transverse perturbations and a convergent, not necessarily absolutely convergent, Perron drift. The limiting characteristic function admits a scalar infinite-product representation in a neighbourhood of the origin and a global matrix-product representation at arbitrary frequencies.

Let $D_n$ be the dihedral group of order $2n$. Consider a continuous-time random walk on $D_n$ driven by arbitrary symmetric rates whose support generates $D_n$. For $p\in[1,\infty]$, we say the pair $(D_n,p)$ is rate-monotonic if for each fixed time $t$, the $\ell^p$-distance between the random walk's distribution at time $t$ and the uniform distribution is monotonically decreasing as a function of the rates. Lyons and White proved that $(D_n,2)$ and $(D_n,\infty)$ are rate-monotonic. Somewhat counterintuitively, they found several pairs $(D_n,p)$ with ${p\in[1,1.997]\cup[2.001,3.999]\cup[4.001,5.995]}$ that are not rate-monotonic, and they asked whether any such pairs exist with $p=4$ or $p=6$. We resolve their question, proving that $(D_n,2m)$ is rate-monotonic for all positive integers $m$ and $n$. In fact, we prove a generalization of this result to a broader family of groups that includes generalized dihedral groups, dicyclic groups, and generalized quaternion groups. In the other direction, we prove that for every real $p\geq 1$ that is not an even integer, there exists a positive integer $n$ such that $(D_n,p)$ is not rate-monotonic. The results of this paper were formally verified in Lean by AxiomProver assuming standard literature.

Jiang, Lee, and Vempala conjectured that if $X,Y\in\mathbb{R}^n$ are independent isotropic log-concave random vectors, then $W_2(L(\langle X,Y\rangle),N(0,n))$ is bounded by a universal constant. Subject to Theorems 1.2 and 2.5 of arXiv:2607.24164v1, we prove this conjecture and a rectangular bilinear-form extension. For independent isotropic log-concave $X\in\mathbb{R}^m$, $Y\in\mathbb{R}^n$, and nonzero $B\in\mathbb{R}^{m\times n}$, put \[ r_4(B)=\frac{(\operatorname{Tr}(B^\top B))^2} {\operatorname{Tr}((B^\top B)^2)}. \] We construct a nonnegative scalar Stein kernel for $X^\top B Y/\|B\|_F$ whose squared $L^2$ discrepancy is at most $20/r_4(B)$, and consequently obtain the same bound for squared $2$-Wasserstein distance to $N(0,1)$. The proof develops an exact covariance identity and deficit decomposition for trace observables of moment-map Stein kernels, together with a stability theorem for positive Stein kernels under log-concave approximation. Taking $B=I_n$ yields \[ W_2^2\left(L\left(\frac{\langle X,Y\rangle}{\sqrt{n}}\right),N(0,1)\right)\leq\frac{20}{n}, \] which is the Jiang–Lee–Vempala conjecture.

We study the microscopic spectrum at the origin of ratios of independent complex Girko matrices. Under bounded-density and finite-moment assumptions, together with circular second-moment matching, we prove compact-uniform convergence of the normalized determinants to a random entire function constructed from the complex hard edge and an independent Ginibre array. The zero divisor of this limiting determinant field has the law of the infinite complex Ginibre process. Consequently, the first finitely many smallest and largest eigenvalue moduli, the inner and outer spectral radii, and eigenvalue counts in fixed bounded sets have universal limits depending only on the second moments. This answers the second-moment question posed by Chafaï, García-Zelada, and Xu for the spectral radii. We also obtain a multivariate determinant field for finitely many perturbation directions and prove the uniform hard-edge comparison needed to extend the conclusions to triangular arrays of atom laws.

We study a class of oriented annular planar maps which are closely related to bipolar-oriented maps. We prove that such maps can be encoded by a 2D lattice walk which converges to a 2D Brownian path in the scaling limit. We further prove a mating-of-trees result for the annulus, where the aforementioned Brownian path and a generalization encodes an LQG annulus decorated by a counterclockwise space-filling SLE loop with parameter $\gamma=16\kappa^{-1/2}\in(0,2)$.

We prove that the normalized $k$-th moment of the Laguerre unitary ensemble, as a function of the dimension, is decreasing for fixed $k\in(0,1)\cup (2,\infty)$ and increasing for fixed $k\in(1,2).$ As a corollary, this implies that the average singular value of complex Gaussian random matrix decreases with dimension. Conjectured by Bandeira, Kennedy, and Singer (Math. Program., 2016), this property was recently established by Hutnik, for non-negative integer shape parameter $\alpha$. Our result extends this monotonicity to real $\alpha\geq 0$.

In this paper, we consider Robbins' problem, which is a full information variant of the well-known secretary selection problem. In this version of the problem, the goal is to minimize the expected rank of the selected candidate among $n$ that are interviewed sequentially, and a decision to select or not the $m^{th}$ candidate needs to be taken right after the interview (so without seeing the last $n-m$ candidates and without recall). We first show how to model instances of Robbins' problem as infinite Markov Decision Processes (MDPs). Then we propose several finite-state abstractions of these MDPs that allow us to approximate the value of the problem for fixed $n$. While it is known that the full memory of past candidates' values is necessary for optimal expected rank minimization, making the analysis of the problem challenging, we highlight simple memory structures that are sufficient for obtaining near-optimal selection strategies. Additionally, we provide approximate values for Robbins' problem for numbers of candidates $n$ up to 100 for which no good approximations were previously known (the exact value is only known for instances where $n \leq 4$ and numerical approximations were for small values of $n$ not exceeding one digit), for all $n : 5 \leq n \leq 100$, we give better approximation than what was previously known.

We consider the distribution-constrained optimal stopping problem introduced by Bayraktar and Miller (Mathematical Finance, 2019) and Beiglbock et al. (PTRF, 2018). Motivated by the multi-marginal Skorokhod embedding problem (SEP) and applications in financial mathematics, we generalize the Root-type solution to the multi-marginal case. The key difficulty is that the associated stopping barriers need not be ordered, so the problem in general cannot be reduced to a sequence of one-marginal problems. First, we give a probabilistic characterization in terms of sequential optimal stopping for an auxiliary time-reversed process, in the same spirit as Cox et al. (PTRF, 2019). Then, we prove the optimality of this construction by martingale arguments for a general class of reward functions, including multi-marginal versions of all examples in Beiglbock et al. (PTRF, 2018) as special cases.

We establish a sharp phase transition for fitting random vectors by an ellipsoid. The random vectors have independent subgaussian coordinates with mean zero, variance one, and a common fourth moment, and the number of vectors is proportional to the square of the dimension. We identify an explicit satisfiability threshold such that, with high probability, a positive definite ellipsoid passes through every data point below the threshold, whereas no positive semidefinite fit exists above it. We also determine the optimal squared fitting error throughout the unsatisfiable regime. In particular, the threshold depends on the coordinate distributions only through their common fourth moment, revealing a fourth moment universality phenomenon. For standard Gaussian data the threshold is $1/4$, resolving the ellipsoid fitting conjecture.

We consider the real noncentral sample covariance matrices $\mathcal{W}=YY^\top$ with $Y=A+\Sigma^{1/2}X$. Here $A\in\mathbb{R}^{M\times N}$ is deterministic, $\Sigma\in \mathbb{R}^{M\times M}$ is a deterministic positive definite population covariance matrix and $X\in\mathbb{R}^{M\times N}$ has independent centered entries with variance $N^{-1}$. We prove local laws near regular right edges down to optimal spectral scales without requiring the commutativity of $AA^\top$ and $\Sigma$. As a consequence, we obtain optimal eigenvalue rigidity at the rightmost regular edge and delocalization of the corresponding left and right singular vectors. We also show that, with high probability, there are no eigenvalues in the adjacent spectral gap beyond the optimal $N^{-2/3}$ edge scale, up to an arbitrarily small $N^\varepsilon$ loss. Finally, we establish edge universality at the rightmost regular edge: after centering and scaling, the largest eigenvalue converges to the Tracy–Widom distribution. The main technical ingredient is a stability analysis of the matrix Dyson equation (MDE) associated with the linearization of $Y$, whose self-energy operator does not satisfy the flatness condition of the general MDE theory. Exploiting the special block structure, we reduce the stability analysis exactly to a two-dimensional operator. This reduction yields regularity of the spectral density and square-root behavior at regular right edges, together with sharp stability bounds near such edges.

In this paper we investigate stochastic heat equations driven by multiplicative noise on moving domains. We establish the well-posedness within a nonhomogeneous variational framework. Furthermore, by combining stochastic De Giorgi iteration with Dirichlet parabolic estimates, we obtain the Hölder regularity of the solutions.

We model equity markets using geometric Brownian particles entering and exiting at rank-dependent intensities. In the many-firm limit, the capital distribution converges to the solution of a reaction-diffusion equation with reaction term built from the intensities. Calibrated on CRSP data, the reaction term is bistable, and the long-run distribution is a traveling wave: we prove existence, uniqueness, and, for constant coefficients, exponential relaxation. Turnover, not drift, stabilizes the calibrated market. With measured volatility, the wave tracks the empirical capital distribution in every decade, determines the capitalization growth of diversity-weighted portfolios, and places the market just inside the boundary of the diverse phase. Turnover reclaims most of what rebalancing gains.

We develop $L^1$ bounds for the difference between a test function of a random sum and a standard normal random variable, where the summands are assumed to be independent but not necessarily identically distributed. The bounds are obtained through a new version of the approximate zero bias transformation specifically developed for random sums. Although the identical distribution assumption is relaxed, the bounds are of order $1/\sqrt{n}$, matching the order of existing bounds in the literature under the same distributional assumption on the number of summands. The main results are then applied to three real-world settings: random sums obtained from simple random sampling with outliers, total insurance claims, and generative AI response times.

Deterministic source-sink models are widely used to assess the long-term evolution, capacity, and sustainability of the low Earth orbit (LEO) environment. These models propagate shell-averaged populations through ordinary differential equations (ODEs), relying on individual collision, disposal, and decay events to average out within sufficiently large altitude shells. As constellation traffic is increasingly organized into kilometer- and sub-kilometer-scale shells, this averaging assumption becomes strained. We formulate the multi-shell, multi-species LEO environment as a Markov jump process and, from this formulation, recover the conventional source-sink ODE as a large-volume limit and derive a stochastic differential equation (SDE) approximation whose fluctuations scale as V^{-1/2} in the per-shell volume V. The diffusion approximation is validated against an exact discrete-event simulation of the underlying jump process. Sweeping V at fixed spatial density over the 450-800 km band, we find that the two descriptions agree at shell volumes comparable to those used in established source-sink models but diverge as shells narrow. At the finest shell volume considered, the mean debris population reaches roughly 4.5 times the ODE prediction, with several realizations undergoing runaway growth absent from the ODE trajectory. This departure is driven by nonlinear collision terms, through which population variance and covariance raise expected collision rates, generating further debris and reinforcing the collision-debris feedback. Narrow-shell stochasticity thus both widens the outcome distribution and shifts the expected trajectory. Because the stochastic model shares its parameterization with the deterministic one, it provides a scale-consistent extension of existing source-sink models for evaluating shell configuration, collision risk, and long-term LEO sustainability.

Motivated by the recent introduction of chromatic persistent homology, we generalize the Euclidean minimum spanning tree (EMST) for $n$ points in $\mathbb{R}^2$ to the lunar EMST for the case in which the points come in $s+1$ colors. Calling the intersection of $s+1$ disks of radius $r$ centered at points with pairwise different colors a lune, the generalized EMST reflects the history of the union of lunes as $r$ goes from $0$ to $\infty$, and its cost is twice the difference between the radii when the arcs and nodes of the tree are formed. If the points are chosen uniformly at random in $[0,1]^2$ and colored randomly, the expected cost converges to some constant (that depends on $s$) times $\sqrt{n}$, as $n$ goes to infinity. The main contribution of this paper is a proof that this constant exists, however similar to the case of the classic EMST, its precise value remains elusive.

We study stochastic hypoviscous Navier–Stokes equations on the torus with dissipation $(-\Delta)^\gamma$ for $\gamma \in (\frac{1}{2},1]$ and multiplicative noise. Relying on stochastic maximal regularity results for the linear equation, we establish local well-posedness for this problem in arbitrary dimensions with initial data in a range of scaling-critical Besov spaces. With the aid of $L^q$-energy estimates for the vorticity equation, we also prove global well-posedness of the stochastic hypoviscous Navier–Stokes equation in 2D with linear multiplicative noise.

This expository note is about simulating a target distribution $Q$ from observations of a proposal distribution $P$. In the model suggested by Harsha, Jain, McAllester and Radhakrishnan, we observe an infinite sequence of i.i.d. samples $X_1,X_2,\ldots$ from $P$. The goal is to find some index $I \in \{1,2,\ldots\}$ such that $X_I$ is distributed like $Q$ while minimizing $\mathbb{E} \log I$. Following the Poisson-race approach developed by Maddison, Li and El Gamal, and others, this note shows that if $D(Q||P) < \infty$ then there is a $P$ to $Q$ simulator $I$ such that $\mathbb{E} [\log I] \leq D(Q||P) + 1.45 \|Q-P\|_1$. In the other direction, for every $P$ to $Q$ simulator $I$, the cost is at least $\mathbb{E}[\log I] \geq \frac{1}{2} \max \{ D(Q||P), \|Q-P\|_1 \}$.

In this paper, we establish a small-time annealed path large deviation principle for one-dimensional diffusions in a random environment associated with the generator ${\mathcal L}_W f(x)=e^{-\rho(x,W)}(e^{a(x,W)}f'(x))'$. The coefficients $\{\rho(x,\cdot):x\in\mathbb R\}$ and $\{a(x,\cdot):x\in\mathbb R\}$ are random. We assume that for each fixed realization of the environment, $\rho$ and $a$ are continuous and locally exponentially integrable, and that the support of the associated intrinsic coordinates is compact and non-collapsing. This framework includes the extensively studied Brox diffusion $dX_t=dB_t-\frac12\dot W(X_t)\,dt$, where $B$ is a standard Brownian motion and $W$ is an independent two-sided Brownian motion representing the environment. The Itô–McKean representation of the diffusions and the estimates of the first exit probabilities derived via Moser iteration play a crucial role.

We study moderate deviations for the largest eigenvalue of the randomly deformed Gaussian unitary ensemble introduced by Johansson (Probab. Theory Relat. Fields, 138: 75–112, 2007). In the fixed-coupling regime, the rescaled largest eigenvalue converges to the convolution of the Tracy–Widom law and a Gaussian law arising from the random displacement of the spectral edge. We derive sharp logarithmic asymptotics for the right and left tails on growing Airy scales and obtain the corresponding phase diagram. The two tails have different transition scales. At criticality, the rate functions are nontrivial infimal convolutions of the Tracy–Widom and Gaussian rate functions. We establish a trace-norm Airy approximation that is uniform over typical diagonal configurations on a logarithmic window. A conditional convolution argument then combines the resulting tail estimates with the Gaussian moderate deviations of the edge displacement.