Papers

We prove a quenched functional central limit theorem for the random conductance model with ergodic translation-invariant, strictly positive nearest-neighbor conductances, assuming only finite first moments of the conductances and their inverses. This settles an open problem by reaching the critical first-moment threshold, which is sharp for a certain class of integrability assumptions. A key ingredient of the proof is a new Sobolev inequality that seems to be missing from the literature.

Locally commuting multiple radial Schramm-Loewner evolutions ($\mathrm{SLE}_{\kappa}$) are encoded by partition functions satisfying the radial Belavin-Polyakov-Zamolodchikov (BPZ) equations and a conformal Ward identity with spectral parameters $\lambda,\nu\in\mathbb{R}$. For $\kappa>0$ and $\lambda\in\mathbb{R}$, we show that the solution space of the radial BPZ system has dimension $2^n$, where $n$ is the number of variables. We then determine all admissible Ward parameters $\nu$ and the exact dimensions of the subspaces selected by the conformal Ward identity, covering both generic and degenerate cases. The classification reveals a parity difference: nonzero rotation-invariant solutions exist for every $\lambda$ when $n$ is even, but only at finitely many exceptional values when $n$ is odd. When $0<\kappa\leq4$, $\lambda>0$ for odd $n$ or $\lambda>-3/2$ for even $n$, we construct a basis of positive solutions using multiple SLE, providing global realizations of the locally commuting SLEs. At $\kappa=4$, we identify a family of explicit solutions as partition functions for level lines of a Gaussian free field with suitable boundary data and interior singularities.

Talagrand asked whether the Rademacher cotype of an operator is universally controlled by the maximum of its Gaussian cotype and its $(q,1)$-summing norm. We give a negative answer already for $q=2$. The counterexample was discovered by ChatGPT (GPT-5.6).

We prove that a path maximizes the expected range of a uniformly chosen graph homomorphism into the integers, with one vertex pinned at zero, among all connected bipartite graphs of the same order. This establishes the expectation form of the Benjamini–Häggström–Mossel conjecture. The proof restricts and rescales a homomorphism on each bipartition class, then contracts the edges on which the resulting height function is constant. A quantitative estimate for the rank of these zero edges compensates for a parity term in the expected range of a simple random walk, allowing an induction on the number of vertices. We then prove that the BHM inequality implies the Loebl–Ne\v set\v ril–Reed inequality for uniformly chosen integer 1-Lipschitz functions on arbitrary connected graphs, and hence obtain the LNR conjecture as a corollary of BHM. The proof was obtained through interaction with OpenAI GPT-6 Astra and verified by the author. The main results have also been formalized and checked in Lean~4.

We give an efficient algorithm with improved algorithmic guarantees for the (offline) Beck–Fiala problem when the sets have bounded size. Let $A$ be an arbitrary matrix $A\in\{0,1\}^{m\times n}$ with at most $d$ ones per column and at most $s$ ones per row. Let $\log^*$ denote the iterated logarithm and $\ell_j$ denote the $j$-fold composition of log. Assume $s\le\exp(O(\sqrt d))$. We provide an efficient algorithm that, for arbitrary sparsity $d$, gives $O(\sqrt d(1+\log^*n))$ discrepancy. Moreover, if $d\ge\ell_j(n)$ for a fixed integer $j\ge1$, the algorithm gives $O_j(\sqrt d)$ discrepancy. The proof is a bootstrapping scheme using the Bansal-Jiang algorithm.

Suppose we observe the first $n$ points of a sequence of random variables having length $n+1$, and wish to estimate a functional of the unobserved final point and the empirical measure of the $n$ observed training points. Such next-token functionals include the probability that the next token is novel (also known as the surprise probability), the tail probability of the minimum distance between the next token and training points, and the test error of a classifier trained on the observed points. All of these quantities are classically estimated by the leave-one-out method, which is inconsistent under temporal dependence. We propose a leave-a-window-out estimator, which deletes a window of length $\tau$ after each index before forming the empirical measure and reduces to leave-one-out at $\tau = 1$. Under natural assumptions, we show that the error of our estimator decays at a parametric rate for any stationary $\beta$-mixing process that also admits a Marton coupling. Our results thus cover several natural functionals on a large class of stochastic processes. We complement these upper bounds with a sharp minimax lower bound for estimating the surprise probability on mixing Markov chains. Simulations on Markov chains, moving-average processes, and autoregressive processes show that our estimator succeeds in many scenarios where leave-one-out and add-constant baselines fail.

Local-to-global techniques for establishing spectral gaps have played a central role in the modern theory of Markov chain mixing times and the theory of high-dimensional expanders. One of the most striking results in this burgeoning literature is that a spectral gap for the global down-up walk on the facets of a pure simplicial complex can be reduced to sufficiently strong spectral expansion of just the codimension-2 links of the complex, a phenomenon colloquially referred to as "trickle-down". These types of theorems have had many important applications, including rapid mixing of the exchange walk on the bases of any matroid. In this primarily expository article, we give streamlined proofs of two such theorems in the literature, one by Oppenheim (2018) and one by Leake and Oveis Gharan (2025), via an integrated Bochner method. Moreover, in the latter setting, we quantitatively strengthen the dependence of the global spectral gap on the dimension of the complex and the spectral influence, resolving an open question of Leake and Oveis Gharan. Disclaimer: The proofs were developed through a couple of rounds of interaction with GPT-5.6 Sol Ultra. We later discovered that Guo and Zhang (2026) had independently proven the same strengthening of the trickle-down theorem of Leake and Oveis Gharan using an extremely similar argument, also found by GPT-5.6 Sol Ultra. The focus of their paper is the complexity of approximating the partition function of spin systems on planar graphs, not on the trickle-down phenomenon itself. In contrast, our motivation is primarily expository, and we hope to bring Bochner-type methods and their connections with the trickle-down phenomenon to the attention of a wider community of researchers.

We define a rank-two extension $\mathcal{H}^{(2)}$ of the spherical $\mathcal{H}$-transform, together with its associated transform $\mathcal{R}^{(2)}$. For sums, both transforms are additive. The transform $\mathcal{R}^{(2)}$ yields a $4\times4$ formula for the average product of two hermitized resolvents, and hence for off-diagonal eigenvector overlaps, recovering the Ginibre, elliptic, and bi-invariant formulas. We also treat correlated pairs of matrices, and obtain overlaps between their eigenvectors or singular vectors.

We prove the existence of a McKean–Vlasov stochastic process with jumps associated to the nonlinear parabolic equation $\partial_t u = \Delta_p u + \Delta_p^s u$ in $\R^N\times(0,\infty)$, where $\Delta_p$ is the $p$-Laplacian and $\Delta_p^s$ is the fractional $p$-Laplacian. The algorithm used is the following : first, after proving the existence of a solution for the PDE presented earlier, we rewrite it as a nonlinear Fokker-Planck-Kolmogorov equation whose solution-measure is guaranted when $p\ge4$. Then we solve the martingale problem associated to our FPKE via a new nonlinear supersition principle. Finally, thanks to the martingale solution obtained, we derive the existence of a weak solution for the McKean-Vlasov's type SDE with jumps whose infinitesimal generator is a << hybrid version >> of the operator $\Delta_p+\Delta_p^s$.

We study the maximal multiplicative persistence of $k$-cycles in random Čech complexes. Let $f:\mathbb{R}^d\to\mathbb{R}$ be a probability density function, let ${P}_n$ be a Poisson process with intensity $nf$, and let $\Pi_{k,n}$ denote the largest death-to-birth ratio among all non-essential $k$-cycles ($1\le k \le d-1$). For the uniform distribution in the unit hypercube, it was proved in [9] that $\Pi_{k,n} = \Theta\left( \left(\frac{\log n}{\log\log n}\right)^{1/k}\right)$. In this paper we sharpen and extend this result to a law of large numbers, for a broad class of distributions. Most significantly, we show that the limiting constant depends only on $d$ and $k$, and not on the probability density $f$. Thus the extreme value of multiplicative persistence exhibits a universality phenomenon. We show that the limiting constant is determined by the asymptotic covering density of the $k$-dimensional sphere. Our proof identifies the geometric mechanism underlying maximal cycles, a persistent isoperimetric inequality, which gives sharp bounds on the number of points needed to generate a highly persistent cycle. By combining covering-density estimates with isoperimetric inequalities, we show that this minimum is asymptotically attained by efficient coverings of a $k$-sphere. A key ingredient is a geometric measure theory argument that uses compactness to relate discrete covering counts to the volume of a limiting cycle.

Solvency games are a gambling problem on infinite-state Markov decision processes in which the state $n \in \mathbb{N}$ represents an investor's fortune. In every round, the investor chooses an action from a finite action set, and every action yields a distribution over integer-valued gains in an interval $\{-\ell,\ldots,m\}$. The risk-averse investor wants to minimise the probability of eventual ruin (reaching a fortune $\le 0$). It was shown in [Berger et al.] that memoryless deterministic optimal strategies exist, but they are not eventually constant in general. Even in the special case of gains in $\{-2,\ldots,1\}$, the optimal strategy may need to make use of two different actions at arbitrarily high fortunes. We show that optimal strategies in solvency games need not be ultimately periodic in general (thus disproving a 2012 conjecture of Kučera). Already in the case of gains in $\{-3,\ldots,1\}$, it is possible for the optimal strategy to be unique but aperiodic. For gains in $\{-2,\ldots,1\}$, there always exists an ultimately periodic optimal strategy whose tail is constant or alternates between two actions. Finally, we show that the optimal strategy is computable if it is unique. Moreover, (some) optimal strategy can always be computed in the case of gains in $\{-\ell,\ldots,1\}$ for any $\ell \in \mathbb{N}$. Computability in the general case however remains open.

Given red and blue points drawn from Poisson point processes of equal intensities, we match them by locally minimizing a cost function. In our paper, we identify all possible measurable cost functions on $\mathbb{R}^d$ such that the resulting matching is scale invariant. Our cost functions can be asymmetrical, which we refer to as unbalanced cost functions. The investigation is motivated by arXiv:2012.07129 where the authors consider scale invariant cost functions with added regularity assumptions. Furthermore, we discuss how different the unbalanced matchings are compared to these more regular cost functions for points on $\mathbb{R}$.

Consider a sequence of i.i.d.~trials $X=\{X_1, X_2, \ldots, X_n\}$ where $p(X_i=j)={1}/{d}; j=1,2,\ldots, d$, or more generally $p(X_i=j)=p_j; \sum_{1\le j\le d}p_j=1$. We consider the variable $D$ that counts the number of distinct substrings of all lengths,$1\le k\le n$ in $X$ and prove results concerning $E(D)$.

We provide a tractable sufficient condition for strict comparison for solutions of the one-dimensional stochastic differential equation \[ d X_t^x=\sigma(X_t^x)\,d B_t, \quad X_0^x=x\in I=(\ell,r), \] for $\sigma>0$ and continuous. Our proof is based on a two-dimensional Lyapunov argument, which allows us to prove strict comparison for some coefficients in $W^{1,p}_{\text{loc}}(I)$, $1\le p<2$. We illustrate using $\sigma(x):=1+|x|^\beta$ for $x\in\mathbb{R}$, $\beta \in (0,1)$, and show that strict comparison holds if and only if $\beta\in[\frac12,1)$. We give examples showing that neither $\sigma\in W^{1,p}_{\text{loc}}(\mathbb R)$, $p\in(1,2)$, nor $\sigma\in C^\beta(\mathbb R)$, $\beta\in[\frac12,1)$, is sufficient for strict comparison, even when combined with boundedness, uniform ellipticity, global strong existence, and pathwise uniqueness.

According to the classical Skitovich–Darmois theorem, the Gaussian distribution on the real line is characterized by the independence of two linear forms of a finite number of independent random variables $\xi_j$. This theorem has been extended in various directions. In particular, S.G. Ghurye and I. Olkin established an analogous result for the case where $\xi_j$ are $n$-dimensional independent random vectors and the coefficients of the linear forms are invertible $n\times n$ matrices. Subsequently, A.A. Zinger and later I.A. Ibragimov investigated linear forms of an infinite sequence of $n$-dimensional independent random vectors. In the present paper, we investigate, for the first time, linear forms of a sequence of independent random variables taking values in a second-countable locally compact Abelian group $X$ under either of the following conditions: $X$ contains no nontrivial compact subgroups, or $X$ contains no subgroups topologically isomorphic to the circle group and has a finite topological automorphism group. Furthermore, we investigate the case where the independent random variables take values in an $\mathbf{a}$-adic solenoid $\Sigma_{\mathbf{a}}$. In all these settings, the coefficients of the linear forms are topological automorphisms of the corresponding group.

We study the gradient-flow training dynamics of quantum physics-informed neural networks (QPINNs) for the solution of second-order elliptic partial differential equations with Dirichlet boundary conditions. We consider parameterized quantum circuits as function approximators and analyze their overparameterized regime through the lens of the neural tangent kernel (NTK). Our contribution is a nonasymptotic lazy-training theory for QPINNs and their variational formulation: we prove that, for sufficiently large circuit width, the nonlinear gradient flow is quantitatively approximated by a linearized NTK model, with explicit bounds depending on the number of qubits, circuit depth, circuit light-cone geometry, and the dimension of the domain of the solution to the PDE.

We obtain upper and lower bounds on the total variation distance between two product Bernoulli measures. The bounds are efficiently computable and match up to an absolute constant factor.

We study a rank-one spiked complex Ginibre ensemble A_N = X_N + sigma u u^*, where X_N is an N by N complex Ginibre matrix with independent entries distributed as CN(0,1/N), u is a deterministic unit vector, and sigma is a nonnegative spike strength. We focus on the maximum real part of the eigenvalues and its behavior near the non-Hermitian BBP transition. For every fixed subcritical spike strength sigma < 1, we show that the maximum real part has the same Gumbel limit as in the unspiked ensemble, after the same centering and scaling. We also derive an explicit convergence-rate correction of order 1/log N for the unspiked Gumbel law. At the critical point sigma = 1, we establish an upper bound for the rightmost eigenvalue at the N^{-1/4} scale using isotropic resolvent estimates and a net argument.

We prove Chvátal's conjecture, posed in 1972: every hereditary family of subsets of a finite set has a largest intersecting subfamily that is a star. More generally, we prove a sharp correlation inequality for increasing Boolean functions $f,g:\{0,1\}^n\to\{0,1\}$. Writing $g^*(x)=1-g(1-x)$, we show that $$ \sum_{\varnothing\ne S\subseteq[n]}\hat{g}(S)^2\max_{i\in S}\mathrm{Inf}_i[f]\le\frac{2\mathrm{Cov}(f,g)\mathrm{Cov}(f,g^*)}{\mathrm{Cov}(f,g)+\mathrm{Cov}(f,g^*)}. $$ When $g$ is antipodal, that is, $g=g^*$, this yields $\mathrm{Cov}(f,g)\ge\frac{1}{4}\min_{i\in[n]}\mathrm{Inf}_i[f]$, the correlation formulation of Chvátal's conjecture due to Friedgut, Kahn, Kalai and Keller.

In this paper we affirmatively resolve the singularity conjecture for finitely supported non-degenerate random walks on cocompact Fuchsian groups. The method we use is based on constructing a pair of geodesic currents, one for the Lebesgue measure and one for the random walk measure, and checking that they are different by using Fourier analysis on the boundary.

We study robust bond valuation with endogenous short-rate feedback under volatility uncertainty. Within the $G$-expectation framework, the dependence of the short rate on the bond price yields a nonlinear fixed-point problem, represented by a quadratic $G$-BSDE for the logarithmic price. Under suitable assumptions, we establish existence, uniqueness, comparison, and stability for bounded finite-horizon solutions. An additional strict monotonicity condition yields a unique bounded infinite-horizon solution and exponential convergence of finite-horizon approximations on compact time intervals. We apply these results to inverse short-rate design, constructing discount-rate coefficients that reproduce admissible smooth bond-price targets at a fixed maturity. For long maturities, we construct feedback rules under which the compensated logarithmic price converges exponentially to a prescribed bounded state-dependent profile, while the asymptotic yield equals a specified target.

We study an uncountable system of McKean-Vlasov SDEs with coefficients of Nemytskii-type which are driven by a family of essentially pairwise independent Wiener processes. These SDEs interact through a Graphon kernel by means of their one-dimensional time marginal law densities evaluated in the spatial coordinate. We prove the existence and uniqueness of probabilistically weak solutions to such SDEs under mild conditions on the drift coefficient and the Graphon kernel. Furthermore, we prove that these solutions are, in fact, probabilistically strong by essentially proving a (restricted) Yamada–Watanabe theorem on Fubini extension spaces. For the associated system of nonlinear Fokker-Planck equations, we prove a new uniqueness result where we can allow for density dependent diffusion coefficients of Nemytskii-type.

Kernel methods, and Gaussian Processes (GPs) in particular, require a Hilbertian distance measure—one whose square is conditionally negative definite (CND)—to guarantee positive semi-definiteness (PSD) of the kernel matrix; a condition that fails for many natural input spaces, including smooth manifolds and spaces of probability distributions. We propose the Sparse Landmark Embedding (SLE) kernel, which eliminates this requirement entirely. Each input is embedded into a sparse feature vector via compactly supported bump functions centered at all |D| training points; applying any standard PSD kernel in this embedding space yields a kernel that is provably PSD for arbitrary distance measures. The compact support automatically controls embedding sparsity, keeping kernel matrices well-conditioned and computationally tractable despite the high ambient dimension. We provide theoretical guarantees on PSD, sparsity, stability, and universal approximation, and demonstrate, using geodesic and Wasserstein distances, that the SLE kernel matches or substantially exceeds domain-specific baselines in both predictive accuracy and uncertainty quantification.

In this article, we study the facilitated exclusion process (FEP) on the $d$-dimensional discrete torus of side length $N$, with $d\ge2$. For Bernoulli initial data with a fixed density $\rho\in(0,1)$, we first prove that, with high probability, the particle-number sector selected by the initial configuration has a unique active recurrent class if $\rho>1-2^{-d}$ and multiple recurrent classes if $\rho<1-2^{-d}$. We then show that, for sufficiently small $\rho$, the process reaches an absorbing state (a singleton recurrent class) within a poly-logarithmic time in $N$ with high probability. In contrast, for $\rho\in(1/2,3/4)$ and even $N$, we establish a polynomial lower bound on the time required to reach the recurrent set. These results provide strong evidence for a double phase transition, analogous to absorbing-state phase transitions in the contact process, activated random walks, and stochastic sandpiles. To our knowledge, these are the first rigorous estimates for absorption and transient times for the FEP in dimensions two and higher.

We consider the boundary trace process of a new class of sticky Brownian motions and study the telegraph signals of the boundary local time. For the fractional telegraph equation \begin{align*} (\sigma/\eta) D^\alpha_t v(t,x) + D^{2\alpha}_t v(t,x) = \frac{\partial^2 v}{\partial x^2}(t,x), \quad t>0,\, x \in \mathbb{R}, \quad \alpha \in (0,1] \end{align*} we provide a probabilistic representation of the solution in anisotropic Sobolev spaces and compare it with well-known representations in the literature. We subsequently discuss the associated processes and provide their pathwise representations. Based on these representations, we introduce a characterization of the local times for a wide class of sticky Brownian motions on $\Omega$ governed by the non-local dynamic boundary condition \begin{align*} \eta D^\alpha_t \varpi(t,x) = - \sigma \partial_n \varpi(t,x), \qquad t>0, \; x \in \partial \Omega \end{align*} where $D^\alpha_t$ denotes the fractional derivative in the Caputo-Džrbašjan sense. We focus mainly on the interval $[a,b]$ to construct a prototype model, and then lay the foundation for the analysis on balls in $\mathbb{R}^d$. The smooth interpolation between wave propagation and diffusion under anomalous dynamics captures the behaviour of the underlying sticky Brownian motion experiencing significantly prolonged trapping times on the boundary. The solution $v$ retains its continuity up to $t=0$ in $H^1(\mathbb{R})$, which corresponds to the mean-square continuity of the telegrapher's process at the initial instant. However, for $t>0$, the severe sticky effect causes the stochastic trajectories to undergo prolonged trapping periods, leading to highly irregular and rough paths for the local time of the sticky Brownian motion. In our construction the Brownian structure appears immediately, rather than only as a hydrodynamic limit.

We introduce an intrinsic notion of p-roughness for continuous paths,for p>1, defined by the uniform convergence of discrete p-energies over all shifted sufficiently fine uniform grids. We prove that this self-averaging property is equivalent to mesoscopic cancellation of the coarse-graining error for discrete p-energy, yielding a characterization that can be verified on a single uniform multiresolution. $p$-roughness refines the finite p-th variation property and implies the invariance of p-th variation across a class of partition sequences. We prove that Brownian motion is almost surely 2-rough and that fractional Brownian motion with Hurst parameter $H$ is almost surely $1/H$-rough. We also derive criteria for p-roughness based on Faber-Schauder coefficients. These results yield partition-robust formulations of higher-order pathwise calculus and energy occupation measures. Finally, we interpret coarse-graining as a renormalization flow for the p-energy and show that the p-roughness class is stable under critical time-amplitude scaling, with linear p-energy profiles as fixed points.

We prove the exponential convergence in relative entropy of the Sinkhorn algorithm for the entropy martingale optimal transport. We assume that the marginals have compact supports and are in strict convex order, the terminal marginal support is convex, and the initial marginal support lies in the relative interior of the terminal marginal support; the reference cost is assumed to be Lipschitz in each variable. Under these assumptions we establish uniform bounds on the dual variables modulo affine gauges. This result allows us to obtain the existence and uniqueness of the optimizer, and its exponential representation in terms of dual variables. We then establish a relative-entropy stability estimate for martingale couplings with different terminal marginals. Proving that the constant in the stability estimate stays uniform throughout the Sinkhorn iteration allows us to establish the exponential convergence of the Sinkhorn algorithm.

We establish a degree-independent bound on spectral independence for log-concave Holant problems on simple graphs. As a corollary, we obtain relaxation-time bounds for Glauber dynamics of $O_\lambda(m)$ for the monomer-dimer model at activity $\lambda$ and $O_{b,\lambda}(m)$ for $b$-matchings at fugacity $\lambda>0$, where $m$ is the number of edges. For uniform $b$-matchings, the relaxation-time bound improves to $O(bm)$. The main proof ideas were found using GPT-5.6 Sol.

Random walks are a standard tool for search problems in which a state can be updated locally and tested for being marked. When updating the state and checking whether it is marked have different costs, two classical strategies optimize different parts of the cost: checking after every step is optimal in the number of updates, while repeatedly checking only after mixing is optimal in the number of checks. For a single marked state $m$ and a walk started from its stationary distribution $\pi$, Dohotaru and Høyer stated that both guarantees can be matched simultaneously, for a walk that checks after blocks of a fixed length; their argument is sketched through quantum walks, and they observe that they know of no classical proof. We give a short and self-contained classical proof of such a tradeoff, for arbitrary irreducible Markov chains. The algorithm replaces the original transition matrix $P$ by the averaged walk $ \overline P_\tau = \frac{1}{\tau}\sum_{k=1}^{\tau} P^k, $ where $\tau$ is of order $\pi(m)HT(m)$. Using a coupling with the original walk and Kac's lemma, we prove directly that the averaged walk hits the marked state in $O(1/\pi(m))$ checks in expectation. The resulting search cost is \[ S + O(HT(m))U + O(1/\pi(m))C \] in expectation, where $S$, $U$, and $C$ denote setup, update, and checking costs.

In this paper, we study the effect of an external random field on the localisation / delocalisation of the long-range Discrete Gaussian Chain (DGC), recently studied by Garban arXiv:2312.04536 and Coquille et al. arXiv:2412.15782. We prove that the model is delocalised at every inverse temperature for any polynomial decay $\alpha > 3/2$. This behaviour differs qualitatively from the one of the disorder-free long-range DGC, for which delocalisation at every temperature occurs only for $\alpha > 2$, similarly to what happens for long-range Ising models as already proved by Aizenman and Wehr. Additionally, we provide a more detailed description of the delocalised phase by identifying the growth exponent and scaling limit of the height at the origin for any decay exponent $\alpha > 3/2$.