Papers

Paths $X \colon [0,T] \to \mathbb R^d$ are traditionally stored in finite memory as time series. Recent research has underscored the benefits of instead representing them as collections of iterated integrals $\{\int_{0 < u_1 < \ldots < u_n < T} \mathrm{d} X_{u_1} \otimes \cdots \otimes \mathrm{d} X_{u_n}\}_{n = 0}^N$. These two encodings can be viewed as the extrema on a two-parameter spectrum of representations of the path as degree-$N$ signatures on $m$ intervals in a partition of $[0,T]$. We ask the question of which such representation takes up the least amount of memory, measured as number of real values needed to store the truncated log-signature, subject to the constraint of it being able to approximate solutions to linear controlled differential equations (CDEs) $\mathrm{d} Y = AY \mathrm{d} X$ with $|A| \leq r$ at accuracy at least $\varepsilon$. Estimating the error in terms of the length of $X$, we find that the optimal representation generally lies strictly in between the two naive choices $N = 1$ or $m = 1$, and derive its asymptotics as $r \to \infty$ and $\varepsilon \to 0^+$. Similar considerations can be made when estimating the error in terms of the $p$-variation norm of $X$: in this regime we prove an error bound of the degree-$N$ Euler scheme for linear CDEs with decay in both $m$ and (factorially) in $N$ with the other arbitrarily fixed. We conclude by setting up the analogous problem for SDEs, with the error measured in $L^2$, and derive a similar $L^2$-Euler error estimate for Itô SDEs with drift. We include an empirical study of the optimisation problem, which we demonstrate for toy examples of $p$-rough paths and for fractional Brownian motion.

We introduce the multispecies version of the harmonic process on a one-dimensional chain in contact with boundary reservoirs. This process is a continuous-time Markov chain where each site can host an unbounded number of colored particles. The symmetric bulk dynamics is put in contact with reservoirs, which inject and remove particles driving the system out-of-equilibrium. The Markov generator of the process is identified with the integrable Hamiltonian of an open rational Heisenberg spin chain of higher rank. We derive the underlying R- and K-matrices in operator form and construct the double-row transfer matrix following Sklyanin. Similar to the monospecies case the R-matrix factorises into two R-operators, each factor corresponding to left and right moving particles. We further define three dual models: an absorbing particle model, a hidden parameter model and a heat conduction model.

We propose a stochastic model of sea spray as a shot noise process attached to a homogeneous Poisson process whose intensity measure is the sea spray generation function, that is to say the average number of droplets of a given radius ejected into the atmosphere per unit area per unit time. We assume the impulse response function defining the shot noise is determined by the trajectory of the center of a spherical droplet whose radius may change with time. We also assume the droplet trajectory is vertical and droplets don't interact with one another. Under these assumptions we show the total mass of airborne sea spray droplets is a stationary stochastic process whose mean and covariance can be written explicitly as functions of the physical parameters. We illustrate the model through examples of increasing physical relevance the most general of which includes gravity, Stokes drag forces, and Langmuir's D-squared law of evaporation. A consequence of this particular example is an estimate of the height of the marine boundary layer at sufficiently large wind speeds.

We derive an explicit formula for the pair correlation function of the Sine$_6$ process in terms of Bessel functions of the first kind. This provides the first single-variable special function representation of the pair correlation function for the bulk limit of a beta-ensemble beyond the classical values of $\beta=1,2,$ and $4$.

We study the extent of universality in the scaling of the height $0$ probability of the stationary Abelian sandpile on lattice approximations of a region $U \subset \mathbb{C}$ with lattice spacing $\varepsilon$ and with open boundary conditions. We show that under certain symmetry assumptions on the lattice, in the scaling limit $\varepsilon \to 0$ this probability at a point $z \in U$ equals $p_{\mathcal{G}}(0) + \varepsilon^2 c_{\mathcal{G}} f_U(z) + O(\varepsilon^3)$, where $p_{\mathcal{G}}(0)$ is the height $0$ probability on the full lattice, $c_{\mathcal{G}} > 0$ is a constant depending only on the lattice, and $f_U$ is a conformally covariant positive real-valued function depending only on $U$. This generalises a result of Brankov, Ivashkevich and Priezzhev (1993), who considered the square lattice and the upper half plane. It also generalizes Example 4.10 in the recent study of the fermionic DGFF of Adame-Carillo and Ruszel (2025). We show via counterexamples that our symmetry assumption cannot be omitted, although conformal covariance may be recovered via an unusual scaling. In particular, the natural analogue of our result fails on general isoradial graphs. Our work is motivated by Jeng, Piroux and Ruelle (2006), who computed the correction terms for all height variables in the special case of the upper half plane in $\mathbb{Z}^2$. They conjectured that their results hold more generally and the present paper is a step in an attempt to make this conjecture rigorous.

We consider the two-species particle system introduced by Stevens (2000) related to the doubly parabolic Keller-Segel equation. It consists of $N$ cells and of a varying number of chemoattractant particles. Cells diffuse in the plane and follow the (mollified) empirical gradient of concentration of chemoattractant. Chemoattractant particles are produced by cells at some constant rate, diffuse and disappear at some constant rate. We show that when the sensitivity of cells to the chemoattractant is small enough, under some rather weak condition on the family of mollifiers, this system approximates the parabolic-parabolic Keller-Segel equation as $N\to \infty$. We also prove that when $N$ is fixed and when the production rate of chemoattractant particles tends to infinity, this system approximates the (non-Markovian) one-species system introduced Talay-Tomašević (2020) and further studied by the authors (2023).

We study additive functionals of stationary Markov chains whose observables depend on a fixed finite block of consecutive states. Such block observables arise naturally in sliding-window statistics, pattern counts, and local dependence analysis. In the independent setting, additive functionals of overlapping finite blocks are known to have a covariance operator with integer spectrum \(0,1,\ldots,k\), and the eigenvalue-one component represents the information first detectable at block length \(k\). We study the corresponding problem when the underlying sequence is a stationary Markov chain. For a fixed block observable \(f(X_t,\ldots,X_{t+k-1})\), we introduce a Hilbert-space decomposition that separates information already contained in shorter consecutive blocks from the genuinely new block-\(k\) component, called the incremental component. We show that this component persists as an eigenvalue-one component under Markovian dependence: on this space the Green–Kubo covariance converges trivially and the covariance operator acts as the identity. More generally, the integers \(1,\ldots,k-1\) are shown to arise hierarchically as eigenvalues of the Green–Kubo operator. In the reversible case, the remaining covariance structure is described through a boundary-corrected one-coordinate marginal spectrum determined by the base transition operator. Reversible two-state chains and Gaussian AR(1) models illustrate the theory through concrete spectral formulas.

This paper addresses statistical inference for stochastic partial differential equations. We study the stochastic Allen-Cahn equation driven by space-time white noise and analyze its mild solution. Our main focus is the asymptotic behavior of the spatial quadratic variation of the solution, for which we establish the exact limiting value. As an application, we develop parameter estimation procedures based on these asymptotic results. We prove that the unique solution can be decomposed as u = X + Y where X denotes the solution of the linear stochastic heat equation and Y accounts for the nonlinear effects. Exploiting a detailed analysis of the heat kernel and its scaling behavior, we derive Hölder continuity properties of Y in both spatial and temporal variables, showing that Y exhibits substantially higher regularity than X. This decomposition and the resulting regularity estimates are key ingredients in the development of parameter estimation procedures based on the asymptotic behavior of quadratic variations of the solution.

The probability-flow ordinary differential equation (PF-ODE) associated with a diffusion process is widely used in score-based generative modeling as a deterministic sampler that reproduces the marginal distributions of the diffusion. The validity of this marginal-matching property depends on the well-posedness of an ordinary differential equation whose velocity field is constructed from the score function of the diffusion. We examine the precise mathematical relation between a diffusion process, the Fokker-Planck equation and the associated PF-ODE under weak regularity assumptions on the drift and score. We establish existence and uniqueness of the marginal density flow as a solution of the Fokker–Planck equation under minimal regularity assumptions. We then study the corresponding Lagrangian problem using the DiPerna-Lions-Ambrosio theory of regular Lagrangian flows. We prove existence, uniqueness and stability of the flow, and show that it transports the initial distribution onto the diffusion marginals, under Sobolev or bounded-variation regularity of the score together with one-sided bounds on the divergence of the probability-flow velocity. We identify sufficient conditions for the required regularity in diffusion models relevant for applications. Our analysis underlines a fundamental distinction between Eulerian and Lagrangian descriptions. We construct a counterexample in which the Fokker–Planck equation has a unique density flow while the associated PF-ODE fails to admit a regular Lagrangian flow from the initial time, demonstrating that uniqueness of the density evolution does not in general imply the existence of a deterministic probability-flow representation. Finally, we derive stability estimates for probability-flow trajectories under learned score approximations. Our findings have implications for the training and deployment of score-based diffusion models.

Under small-ball bounds for the Malliavin determinants of two $\mathbb R^k$-valued Sobolev mappings on a Gaussian space, we estimate the total variation distance between their laws both in terms of the Kantorovich–Rubinstein distance and in terms of the distance between the mappings in the corresponding Sobolev space. In particular, our results yield new total variation distance estimates for distributions of random vectors whose components belong to finite sums of Wiener chaoses, with exponents improved by an asymptotic factor of two. The proof is based on fractional regularity estimates for distributions of Sobolev mappings. Namely, we show that if an $\mathbb R^k$-valued mapping has components in $W^{2,p}(\gamma)$ and the determinant of the corresponding Malliavin matrix satisfies a small-ball bound of order $\varkappa\in(0,1]$, then the law of the mapping has fractional regularity of order \[ \frac{\varkappa}{1+(2k-1)\varkappa p^{-1}}. \] In particular, for large $p$, this gives regularity of order $\varkappa$ up to an $O(p^{-1})$ loss.

We prove a realization theorem for stealthy point processes based on lattice induction, together with a converse in dimension one. Every probability-preserving $\mathbb R^d$-action induced from a full-rank lattice admits a generating Delone cross-section whose Bartlett spectrum vanishes on a neighborhood of the origin. The construction can be chosen so that first-order linear statistics detect a nonzero part of the inducing spectral type. Bernoulli bases yield a nonzero absolutely continuous component, while weakly mixing bases of singular maximal spectral type yield a nonzero singular-continuous component. To our knowledge, these are the first rigorously constructed translation-invariant stealthy point processes on $\mathbb R^d$ with non-pure-point Bartlett spectrum. Conversely, let $\eta$ be an ergodic translation-invariant point process on $\mathbb R$ with positive intensity and local second moments. If \[ \int_{0<|\xi|<1}\frac{1}{\xi^2}\,d\sigma_\eta(\xi)<\infty, \] then its translation action has a nonzero eigenvalue and is lattice-induced. Consequently, an ergodic probability-preserving Borel $\mathbb R$-space is lattice-induced if and only if it admits a generating stealthy Delone cross-section. This should be compared with a theorem of Borichev, Sodin and Weiss stating that a translation-invariant point process on $\mathbb Z$ with proper spectral support is periodic.

We start by surveying known properties of first passage percolation (FPP) geodesics on a Gromov-hyperbolic group $G$. Due to a coalescence phenomenon established in earlier work of the authors, a random tree $T(\xi,\omega)$ consisting of infinite random geodesics to a point $\xi$ on the Gromov boundary $\partial G$ emerges naturally. These trees have a rich random geometry that we explore further in this paper.

We study the asymptotic spectral properties of high-dimensional Spearman correlation matrices for scale-mixture data. We consider observations of the form $x_t=\sigma_t \xi_t \in \mathbb{R}^N,$ where the coordinates of $\xi_t$ are i.i.d.\ and the scalar mixture variable $\sigma_t$ is shared by all coordinates. Under natural symmetry assumptions, the coordinates of $x_t$ are pairwise uncorrelated in both the Pearson and Spearman sense. Nevertheless, they are not independent when the mixture variable is non-degenerate. We show that this higher-order dependence survives the rank transformation and leaves a nontrivial spectral signature. In the proportional regime $N/T\to q\in(0,\infty),$ the empirical spectral distribution of the Spearman correlation matrix converges almost surely to a generalized Marčenko–Pastur law governed by the limiting distribution of an effective rank variance. We also formulate a broader latent-variable extension, which covers, in particular, some scale-mixture models with correlated directional components. We discuss solvable examples and numerical approximations, motivated in part by heavy-tailed data in robust multivariate statistics, econometrics, and finance.

We prove that the local limit of the minimum spanning arborescence in the complete graph (which is an oriented cousin of the minimum spanning tree in the complete graph) is the same as that of the uniform random tree, oriented towards infinity. The latter is known to be the critical Poisson Galton–Watson tree conditioned to survive. This is in sharp contrast with the local limit of the minimum spanning tree, which is known to be different from that of the uniform spanning tree (due to results of Addario–Berry \cite{AddarioBerry2013} and Addario–Berry, Griffiths and Kang \cite{PWIT_local_minimum}). Thus we demonstrate that introducing orientations changes the local geometry of the minimum spanning tree in a non-trivial manner.

This note considers a constrained-degree percolation model in a random environment on the square lattice, where each vertex is assigned a random degree constraint, and edges attempt to open at random times. We prove that whenever an infinite open cluster exists, it is almost surely unique, implying the positivity of the connectivity function in the supercritical regime.

Let $\boldsymbol{X} = (X_1,\ldots,X_n)$ be a centered Gaussian vector, not necessarily nondegenerate. It is proved that, for every $\alpha_1,\ldots,\alpha_n > 0$, \[ \mathsf{E}\left[\prod_{i=1}^n |X_i|^{\alpha_i}\right] \geq \prod_{i=1}^n \mathsf{E}\left[|X_i|^{\alpha_i}\right]. \] When all marginal variances are positive, equality holds if and only if the coordinates are independent. This settles Frenkel's 18-year-old Gaussian product inequality (GPI) conjecture and, in fact, its later strengthening to arbitrary positive exponents. Through Frenkel's hafnian formulation, this result also provides a short proof of the 28-year-old real linear polarization constant conjecture, which was settled this year as a consequence of the strong polarization inequality. The main result leads to the exact real linear polarization constant, a sharp weighted product inequality for real linear functionals, spherical moment bounds, hafnian inequalities for positive-semidefinite matrices together with complete characterizations of their equality cases, logarithmic variance and covariance inequalities, a Rényi total-correlation certificate, and unconditional versions of mixed-sign Gaussian moment bounds previously conditional on the positive-exponent GPI.

Let $X\sim\text{Mult}(N;p_1,\dots,p_n)$ with $p_i>0$ and $\sum_ip_i=1$. We study $\Pr\{X=m\}$ for lattice points near $Np$ from two complementary viewpoints. First, we give an exact integer-mode criterion: the multinomial mode is the Jefferson–D'Hondt divisor apportionment, given when the mode is unique by $m_i=\lfloor (N+d)p_i\rfloor$, $d\in[0,n)$. For $n\ge3$ the mode need not be obtained by rounding each $Np_i$ to a neighbouring integer; the excess $m_i-\lceil Np_i\rceil$ can reach $n-2$, so the two-sided localisation $|m_i-Np_i|<1$ stated in a standard reference is not valid in general. Second, by writing the mass as a gamma quotient with unequal scalings, we derive the complete local expansion of $\Pr\{X=m\}$ in integer powers of $N^{-1}$, with all coefficients in closed Bernoulli-polynomial form. The expansion contains the known $O(N^{-1})$ local limit theorem, reduces to the binomial local mass for $n=2$ and to the central multinomial coefficient in the symmetric case, and admits Cesàro averages of the oscillating coefficients in closed $\zeta$-value form.

We prove two families of distinguished cases of the heat flow conjecture for random matrices of Hall-Ho: elliptic Gaussian sources with planar targets, and deterministic Hermitian sources with targets on the boundary of the covariance disk. In particular, the empirical zero measure of the heat-evolved characteristic polynomial of a complex Ginibre matrix converges almost surely to the semicircle law. The strategy of the proof is to transfer estimates for the corresponding well-understood Gaussian matrix ensembles, through exact heat identities, into control of the zeros of the heat-evolved characteristic polynomials.

We characterize a noisy variant of a Quantum public encryption protocol recently introduced by Malavolta and Walter which demonstrated that the notion of everlasting security can be rigorously formulated for Quantum key distribution after two rounds of interaction between Alice and Bob. To address one possible direction of research that is related to injecting noise in the cryptographic protocol related to Quantum key distribution we formulate arguments for further examining the notion of everlasting security through malleability assumptions on transformations of the ciphertext. Assumptions surrounding malleability were introduced by Maurer and Tackmann for the purposes of comparing how authenticate then encrypt, and encrypt then authenticate, protocols behave through a variety of expressions for the forwarding error, deleting error, and reconstruction probabilities. Such probabilities are put to further use for obtaining connections between the indistinguishability and security threshold for a cryptographic protocol of interests. To further build upon such associations we demonstrate, through an adaptation of the Gentle Measurement Lemma from Quantum information theory, how upper bounds on the trace distance can be used to generalize the negligibility function obtained by Malavolta and Walter in the noiseless setting. Besides the fact that the negligibility function in the noisy setting is related to a higher security threshold it continues to remain of interest to determine whether computations provided in this work for upper bounding the trace distance can be related to other settings that are centered more on game-theoretic approaches.

We investigate the relationship between the space of ends of a unimodular random hyperbolic manifold and the dynamics of its geodesic flow. We show that having two ends of infinite volume implies recurrence, while having infinitely many such ends implies positive drift and entropy. We also provide a transience criterion which applies to deterministic hyperbolic manifolds. Our method relies on studying Delaunay graphs over point processes and on the analytic notion of capacity.

We prove that the vertices of any $n$-vertex graph can be partitioned into pieces of radius $r = O(\log n)$ such that the sum of the sizes of their closed neighborhoods is at most $4n$. This answers a recent question of Bukh and Dubroff and directly yields an improvement to their upper bound on the optimal cover time of the $\epsilon$-steered random walk. We also demonstrate that our bound on $r$ is best possible up to a constant factor for graphs with strong vertex expansion.

This article is primarily expository, but it also contains several new observations and reformulations concerning the Möbius function, probabilistic models, dynamical systems, and the Riemann hypothesis. Its starting question is classical: in what sense can the Möbius function be said to behave randomly? We begin with Denjoy's random-walk heuristic and place it in the context of later work on random multiplicative functions, short intervals, and Möbius pseudorandomness. We then develop two smoothing forms of the classical Mertens criterion for the Riemann hypothesis. The first uses the discrete Laplace transform \[ \Phi(t)=\sum_{n\geq1}\mu(n)e^{-nt} \] and identifies RH with the condition \[ \Phi\in L^p(0,\infty) \qquad\text{for every }1\leq p<2. \] The second uses normalized Möbius Fourier polynomials and local moments on arcs of length comparable with \(1/N\). The paper also revisits the author's earlier criterion for RH in terms of discrete measures, as presented in Broughan's account of analytic equivalents of RH. Denjoy's heuristic is made precise in a simple independent coefficient model, and this model is carefully distinguished from the modern theory of random multiplicative functions. The resulting coefficient space also gives a measure–category contrast: the relevant \(L^p\)-property has full measure but is topologically meagre. A further point of the paper is dynamical: the multiplicative semigroup of positive integers acts naturally on the coefficient space, and the Möbius sequence is a distinguished arithmetic point for this action. The aim throughout is explanatory, while keeping these new observations visible: to show how arithmetic, probability, Fourier analysis, Mellin transforms, and dynamics fit together, and to indicate what modern results add to the older random-walk picture.

A derangetropy operator reweighs a probability density by a fixed profile of its own cumulative distribution function, acting through ranks alone. We prove that these operators are precisely the transformations of absolutely continuous laws equivariant under monotone changes of variable, and that they compose through interval maps, making their dynamics exactly solvable: iteration condenses every law onto its median with a universal Koenigs limit law, the continuous flow is solvable in closed form with the Cauchy family as invariant hyperbolic manifold, and balanced against diffusion the distribution function obeys an overdamped sine-Gordon equation whose unique steady law, the hyperbolic secant, is a globally stable kink. A variational principle selects the canonical kernel, the squared ground state of the Dirichlet Laplacian on the unit interval, whose update is a Bayesian posterior costing exactly one bit for every law, and a unitary lift equips each law with an isospectral Sturm-Liouville system and a quantum carpet: a strong law of large numbers for the critical Sobolev mass, with sharp constant given by Wiener's jump statistic, proves the fractality of the Schrödinger density for arbitrary real bounded-variation data with a jump, previously known only for rational step data. A lift to the dual of the Virasoro algebra, at central charge one, a normalization, identifies laws with disconjugate Hill potentials, tails with conformal weights, and the Cauchy family with the exceptional orbit. Randomized phases yield an exact sampler and a multiplicative chaos, log-correlated in the weakly tempered regime; in several dimensions, dependence stratifies into a conserved interaction, a torsion driven by maximal correlation, and a flat Sinkhorn transport.

We introduce heavy-Wigner random tensors and establish the convergence of a generating family of invariant linear forms in their entries together with a central limit theorem. The relevant group actions arise naturally in the study of tensor eigenvalues. Then, we compute the limiting spectral measure of the matrix obtained by contraction of a heavy-Wigner tensor and we derive two applications. The first one concerns the adjacency tensor of a Erdős-Rényi hypergraph. We identify the local weak limit of the hypergraph and study the limiting spectral distribution of the matrix contraction. The second application concerns Lévy tensors whose entries belong to the domain of attraction of an $\alpha$-stable law.

We present a distinctive approach to parameterizing the risk neutral distribution. Using parsimonious and interpretable parameters, the model provides direct and localized control over the shape of the implied volatility curve. It captures a wide variety of shapes, including those with local concavity. Empirical results demonstrate accurate calibration across a quarter million curves from a two-year Standard and Poor's 500 index option dataset. The fitted parameters exhibit stable patterns across tenors, enabling term structure interpolation and dynamic process construction without static arbitrage.

Majority dynamics is a two-opinion process in which each vertex repeatedly updates to the majority opinion among its neighbors. We study this process on a resampled sparse binary stochastic block model in the assortative regime. At each time step, a graph is sampled from the current opinion partition: vertices with the same opinion are joined with probability $\alpha=a\log N/N$, while vertices with differing opinions are joined with probability $\beta=b\log N/N$, where $a>b>1$. Let $B_t$ and $R_t$ denote the blue and red camps at time $t$. We show that the weighted advantage $\widetilde{\Delta}_t =b|B_t|-a|R_t|$, rather than the unweighted advantage $\Delta_t=|B_t|-|R_t|$ alone, governs the pace to unanimity. Our results, which hold with high probability as \(N\to\infty\), identify three regimes for blue unanimity under the initial blue advantage, i.e., $\Delta_0>0$: constant time, subpolynomial time, and polynomial time. First, when $\widetilde{\Delta}_0 \gtrsim -N/\sqrt{\log N}$, blue unanimity occurs within three updates. Second, when $\widetilde{\Delta}_0 < 0$ and $|\widetilde{\Delta}_0| = o(N)$, blue unanimity occurs within $N^{o(1)}$ updates. Furthermore, when $\widetilde{\Delta}_0 < 0$, $|\widetilde{\Delta}_0| = O(N)$, and $\Delta_0\gg\sqrt{N/\log N}$, blue unanimity still occurs within $N^{I_0+o(1)}$ updates, where \[ I_0= \left(\mathbf{ReLU}\Big(\sqrt{a\frac{|R_0|}{N}}-\sqrt{b\frac{|B_0|}{N}}\Big)\right)^2, \] and $\mathbf{ReLU}(x)=\max\{x,0\}$. Conversely, away from the weighted threshold, when $|B_0|/|R_0|\le a/b-\kappa$ and $\Delta_0>0$, $N^{I_0 - o(1)}$ updates are necessary for blue unanimity. Our analysis relies on detailed estimates for one-vertex flip probabilities in sparse binomial differences, which could be of independent interest.

We introduce a Morse theoretical approach to the construction of the Yang–Mills measure on the space of connections of a compact Riemannian surface. This provides a direct continuous version of this measure which was previously obtained through lattice approximations by Chevyrev in the case of the flat torus and by one of the authors and Nohra for general compact Riemannian surfaces. The starting point is the new notion of a Morse gauge together with the resolution of random cohomological equations associated to Morse–Smale vector fields. This is achieved by improving exponential convergence to equilibrium results for Morse–Smale gradient flows that were obtained by two of the authors in the context of the study of Ruelle spectra and by Jia, Stewart and Sverak in the context of simplified models from fluid mechanics. Combining these random solutions with the data given by the Morse complex, we introduce a free Yang-Mills measure on space of connections and, using classical tools from stochastic differential equations, we show how to make sense of holonomies for random connections along a large class of curves. Finally, by setting a proper conditioning of this free measure through these random holonomies, we define the Yang–Mills measure and we compute its partition function together with the law of random holonomies with respect to this measure, recovering the formulas from the works of Migdal, Witten and Lévy.

The totally asymmetric simple exclusion process (TASEP) with open boundaries is a finite Markov chain describing particles hopping between adjacent sites on a one-dimensional lattice with left and right boundary transitions governed by parameters $\alpha$ and $\beta$. The multispecies TASEP is a higher-rank generalization in which particles have different species. Multiline queues were introduced by Ferrari and Martin (2007) to compute the stationary distribution of the multispecies TASEP on a circle. It has remained an open problem to find a combinatorial formula for the stationary distribution of the multispecies open-boundary TASEP. Using Kirillov–Reshetikhin crystals of type $C$, we construct type $C$ multiline queues and a corresponding Ferrari–Martin pairing algorithm that projects them to TASEP configurations. This yields a combinatorial formula for the stationary distribution of the multispecies open-boundary TASEP for the $\alpha=\beta=1$ specialization.

In this paper, we construct a Lagrange-type basis for the Aomoto space $AO(\mathcal A)$, naturally indexed by the chambers of the hyperplane arrangement $\mathcal A$. The construction relies on a dimension theorem of Orlik and Terao and yields an interpolation formula for elements of $AO(\mathcal A)$. We use this formula to characterize the extremal configurations in the strong polarization inequality as those arising from finite Coxeter reflection systems. We further show that the interpolation formula gives rise to a family of chamber identities, including identities that were central to our earlier proof of the strong polarization problem and the Gaussian product inequality. Finally, we adapt the recent breakthrough of Ouimet and Greaves to prove a generalized Gaussian Product Inequality for completely monotone functions.

For critical Bernoulli bond percolation on $\Z^d$ in the high-dimensional regime, we prove that the rescaled empirical measure of the cluster of the origin converges, in a suitable $\sigma$-finite sense, to the total occupation measure of super-Brownian motion. Combined with a uniform lower mass bound for the critical cluster with respect to the extrinsic (Euclidean) metric, which we also prove and which is of independent interest, the measure convergence further yields the convergence of the rescaled cluster as a compact set, in the Hausdorff metric. As a consequence, we are able to obtain the sharp one-arm asymptotics $r^2\,\bP(0\leftrightarrow \partial B_r)\to \theta_1\in(0,\infty)$, hence refining a result of Kozma and Nachmias.