Papers

We prove two deterministic results for families of distances arising in the study of canonical processes. The first derives an admissible partition scheme from a growth condition. The second gives a representation in terms of parameterized separation trees and compares it with the corresponding majorizing-measure quantities. The main point is that the proofs do not depend on the distribution of the underlying process: once the initial distance and the family of distances are given, no random variables, independence, tail functions, or moment estimates are used. For canonical processes with regular log-concave tails, the assumptions of the abstract results follow from the usual regularity conditions. One direction of the separation-tree estimate also applies to Bernoulli processes without these additional assumptions, and we prove the reverse estimate for bounded convex unconditional index sets. We also give a version of the growth argument for points which, for finite index sets, leads to a recursive construction of admissible partitions and parameterized separation trees.

Given a symmetric simple random walk $(X_n)_{n \ge 0}$, the family of all symmetric simple random walks $(Y_n)_{n \ge 0}$ adapted to the filtration of $(X_n)_{n \ge 0}$ was studied in Collevecchio et al. (2022). In particular, the authors established necessary and sufficient conditions under which the suitably normalized two-dimensional process $((X_n, Y_n))_{n \ge 0}$ converges weakly to a two-dimensional Brownian motion. When this occurs, we say that the random walk $(Y_n)_n$ is normal (with respect to $(X_n)_n$). In this paper, we investigate whether a "randomly selected" $(Y_n)_{n \ge 0}$ is normal. We consider a very general randomization procedure and look at both the quenched and annealed settings.

We prove asymptotic bias bounds for stochastic gradient Langevin dynamics in Wasserstein distance of order two. We assume that the negative log-density is strongly convex with a Lipschitz gradient, and that the stochastic gradient estimator is unbiased with an error satisfying a mean-square Lipschitz condition. The bounds are of order $h$ under a fourth moment assumption on the stochastic gradient error and of order $h^{1/2}$ under only a second moment assumption, where $h$ is the stepsize. A spiked-noise example shows that a second moment assumption alone is insufficient for a bound of order $h$ that is uniform over noise distributions with a fixed variance.

Structural system theory studies which network topologies can sustain a prescribed system property such as controllability or stability. When the topology is itself random, the relevant question becomes probabilistic: how likely is a graph drawn from a stochastic model to sustain the property? Such probabilities measure the abundance and robustness of the property across topologies, and indicate whether systems requiring it can be reliably deployed in uncertain environments. We address this question for asymptotic stability of linear systems in the directed graphon setting. We consider two graph-theoretic properties. The first is $\mathcal N$, and it requires that for every $k\leq n$ some $k$-vertex induced subdigraph of $D$ admits a cycle cover, and the second is $\mathcal S$, which requires that these subdigraphs can be chosen so that their node sets form a nested sequence $V_1\subset\cdots\subset V_n=V(D)$ starting from a single vertex with a loop. We have shown that $\mathcal N$ is necessary and $\mathcal S$ is sufficient for structural stability. We sample $D$ from a directed step-graphon $W$. Our main results give necessary and sufficient conditions for $\Pr(\mathcal N)\to 1$ and $\Pr(\mathcal S)\to 1$ as $n\to\infty$. In more detail, to a step-graphon $W$ with skeleton digraph $S$ on $q$ nodes and concentration vector $x^*$ we associate a cycle polytope $\vec{\mathcal X}(S)\subseteq\Delta_q$. The conditions are then formulated in terms of the position of $x^*$ within $\vec{\mathcal X}(S)$, the dimension of the polytope, the loop density of $W$ and, for $\mathcal S$, an ordering condition on the cycles of the skeleton. Together these results identify, for directed step-graphons, the regime in which a sampled topology is overwhelmingly likely or unlikely to sustain stable dynamics.

We study the average-case matrix discrepancy problem: given independent normalized $d\times d$ Gaussian orthogonal ensemble matrices $A_1,\dots,A_N$ and a fixed margin $\kappa>0$, find signs $\sigma_1,\dots,\sigma_N\in\{-1,1\}$ such that the operator norm of $\sum_{i=1}^N \sigma_i A_i$ is at most $\kappa\sqrt{N}$. Focusing on the proportional regime $N/d^2\to \tau\in(0,\infty)$ as $d\to\infty$ followed by the small-margin limit $\kappa\downarrow 0$, we characterize the density required by stable offline algorithms and by online algorithms. In the offline setting, we construct a polynomial-time recenter-and-round algorithm that is noise-stable and succeeds whenever $\tau=\Omega(\frac{1}{\kappa^2\log(1/\kappa)})$, along with a matching lower bound for all stable algorithms. In the online setting where each sign must be chosen irrevocably upon observing the corresponding matrix, we determine the exact limiting performance of the Frobenius-greedy algorithm, establishing that it succeeds when $\tau>\tau_{\rm FG}(\kappa)\sim \frac{\pi}{4\kappa^2}$, as well as a matching lower bound for all online algorithms by conditioning on a revealed prefix. At the core of our algorithms lies rotational symmetry, which enables us to transfer Frobenius norm control into operator norm guarantees. Together, our results identify the algorithmic phase transition points for random matrix discrepancy: $\Theta(\frac{1}{\kappa^2\log(1/\kappa)})$ for stable offline algorithms and $\Theta(\frac{1}{\kappa^2})$ for online algorithms. Both thresholds lie far above the satisfiability scale $\Theta(\log(1/\kappa))$, as shown by Maillard~\cite{maillard2025}.

We consider the distribution-constrained optimal stopping problem $\sup_{\tau\sim \mu} \mathbb E[B^*_\tau]$, where $\mu$ is a probability distribution on $\mathbb R_+$, and $(B^*_t)$ denotes the running maximum of a standard Brownian motion. This problem was introduced in Beiglbock et al. (PTRF, 2018), where a monotonicity principle is used to establish the optimal stopping time as the hitting time of a specific boundary. In this paper, we characterize this boundary by a variational inequality. In the spirit of Cox et al. (PTRF, 2019), we provide a novel probabilistic representation for the variational inequality as a time-reversed optimal stopping problem. A key ingredient for proving the viscosity solution property and comparison principle is a quantitative estimate near the singular corner of the time-space domain, where the initial and boundary conditions are incompatible. We then prove the optimality of the resulting hitting time through a discrete-time Snell envelope construction and a stability argument for the associated stopping times.

We establish viability conditions for a broad general class of stochastic differential equations formulated within the variational approach,% \[ dX_{t}=A\left( X_{t}\right) dt+B\left( X_{t}\right) dW_{t}. \] The underlying functional space setup is given by a general Gelfand triple $V\subset H\subset V^{\ast},$ where $H$ is a separable Hilbert space in which the stochastic evolution equation (SEE) is considered. Our approach to deriving viability conditions is inspired by the techniques developed by Aubin and Da Prato for finite-dimensional forward stochastic differential equations. We establish necessary and sufficient conditions for the viability of closed random constraint sets in terms of adapted variational tangent (and contingent) sets. Furthermore, we illustrate the applicability of the proposed framework by discussing several important classes of stochastic differential equations that fit naturally within our variational setting.

We study a mixed regular–singular control problem for stochastic evolution equations in a Hilbert space with possibly unbounded random linear operators, a nonconvex regular-control domain, and a state-dependent singular coefficient. The singular control is an adapted nondecreasing càdlàg process whose terminal value need not be bounded. We prove well-posedness and weighted moment estimates for the forward and backward equations, and characterize the second-order adjoint by a conditionally expected operator-valued backward stochastic integral equation. An Itô-type formula for the quadratic form of this adjoint, together with spike and convex variations, yields a second-order Hamiltonian condition for the regular control, as well as nonnegativity and a contact condition for the optional singular Hamiltonian.

The Kannan–Lovász–Simonovits (KLS) conjecture asks whether every isotropic log-concave probability measure on $\mathbb R^n$ has a Cheeger constant bounded below by a universal positive constant. We prove that $\psi_n\le C$ for a universal constant $C>0$, resolving the KLS conjecture. We also prove that $C_P(\mu)\le C'$ for every isotropic log-concave probability measure $\mu$ on $\mathbb R^n$, with a universal constant $C'>0$.

Weak perturbations of the same size can lead to terminal Wasserstein errors of different orders. For infinitely divisible laws smoothed by a Gaussian, we differentiate their distribution functions with respect to variance-weighted Lévy characteristics. This includes distributional directions caused by moving atoms. We obtain first- and second-order expansions that are uniform over Lipschitz test functions. When the first response is nonzero, Gaussian analyticity gives the signed second coefficient of the Wasserstein–1 distance. When the first response vanishes, a finite second displacement moment is enough to obtain a quadratic response. Its coefficient is strictly positive if the displacement dispersion is nonzero. For local balanced remeshing, the error is comparable to the grid-alignment variance. This gives sharp quadratic grid rates for compactly supported densities and nonaligned atomic sequences. We derive metric speed and length along admissible non-atomic curves, and use the response to build a linear program with a proved bound on how much its error exceeds the infimum over the allowed class and consistent quadrature. We also prove a sharp transition as smoothing vanishes and give a multivariate second-order expansion. The appendix gives state-dependent response formulas and further stability estimates.

We study conditional fluctuations of smooth mean-field systems on a torus with endogenous symmetric stable common jumps of index $0<\alpha<1$. A positive-strip estimate makes empirical Poisson acceptance errors negligible at the sampling scale. The conditional laws given the full Poisson master converge in probability to a centered Gaussian kernel in a negative Sobolev path space. In dimension one, quantile coupling preserves this limit for microscopic marks with relative tail remainder $O(x^{-\rho})$, $\rho>\alpha/2$. A quantified second-order tail expansion gives the endogenous rank response: it shifts the conditional Gaussian mean at criticality and, when nonzero, dominates sampling below criticality. We also identify a separate stable-to-Brownian transition for the rank error. Finally, in a translation model, rank and clock couplings give the same microscopic laws. The rank coupling has a functional Gaussian limit. The clock coupling has no diverging additive path normalization at the limiting center.

We consider polynomial processes, i.e., continuous-time Markov processes that preserve polynomials without increasing degree, and augment them with resetting. Resetting means that at the jump times of an indepen- dent renewal process, the polynomial process is instantaneously reset to a new state, which can be chosen deterministically, randomly or in state-dependent fashion. We show that under certain types of resets ('affine resetting') the moments of the resulting process remain tractable and can be calculated by matrix exponentials (for Poissonian reset times) or by a matrix renewal equa- tion (for more general reset times). Our framework encompasses many models that have been considered in the existing literature, such as growth-collapse processes with deterministic growth or with Lévy inflow, and Brownian motion with resets to the origin, to a fixed distribution or by multiplying by a fixed proportion.

Classical central limit theorems (CLTs) for log-partition functions of Ising models on Erdős-Rényi random graphs lead to degenerate limits in the high-temperature regime when self-loops are absent. In this paper, we fill this surprising gap in the more general context of Ising models on inhomogeneous random graphs. In particular, we show that if $G(N,W)$ is an inhomogeneous random graph without self-loops generated by a graphon $W$, then, in the high-temperature regime, the log-partition function of the corresponding Ising model satisfies a CLT on the $N$-scale, with a limiting variance that incorporates contributions from cycles of all orders. This behavior contrasts sharply with that of the corresponding model in which self-loops are allowed, where the analogous Gaussian fluctuations occur on the $\sqrt{N}$-scale, with their asymptotic variance determined entirely by the diagonal profile of $W$. Our analysis yields, as a byproduct of independent interest, a CLT for the log-determinant of the associated resolvent matrix, established by combining the Lindeberg CLT for triangular arrays with suitable bounds on higher-order trace terms.

In this paper, we consider the quasi-copulas and copulas with a prescribed sub-track section, i.e., with given values along the curve segment connecting the points (x0,0) and (1,1-x0) for 0<x0<1. By employing the technique based on the W-ordinal sum, we provide methods for constructing quasi-copulas and copulas whose sub-track section coincides with a given one. Then, we further investigate construction methods for symmetric quasi-copulas and copulas with a prescribed sub-track section. Furthermore, we determine the best-possible bounds for the class of quasi-copulas sharing a common sub-track section, and verify that the lower bound obtained is also the best-possible lower bound for the class of copulas with a common sub-track section. Finally, we examine several properties of the bound (quasi-)copulas with a prescribed sub-track section, including their comparison with bound (quasi-)copulas with a given track section or a sub-diagonal section, as well as their statistical characteristic and concordance order.

For a class of nonlinear Fokker-Planck equations for probability densities, we construct distributional solutions by using an approximation scheme with nonlinear Neumann problems in balls. Moreover, we introduce nonlinear functional inequalities to estimate the convergence rates of the distributional solutions to the stationary solution, with respect to the $p$-variance for $p\in [1,2]$ including the relative entropy $(p=1)$ and the variance $(p=2)$. These nonlinear functional inequalities are then established in both non-degenerate and degenerate settings, so that the main results are applied to a number of typical models with various convergence rates.

We study the wetting transition for a two-dimensional Solid-On-Solid (SOS) interface above a hard wall in the presence of quenched disorder. The interface is represented by a nonnegative integer-valued height function $\phi$, with Hamiltonian $\mathcal H^\omega(\phi)=\beta\sum_{x\sim y}|\phi(x)-\phi(y)|-\sum_x(h+\alpha\omega_x-\lambda(\alpha))\mathbf 1_{\{\phi(x)=0\}}$, where $(\omega_x)_{x\in\mathbb Z^2}$ is an i.i.d. centered field, $\alpha\ge0$ is the disorder strength, and $\lambda(\alpha)=\log\mathbb E[e^{\alpha\omega_0}]$. This normalization is chosen so that the annealed model coincides with the corresponding homogeneous wetting model. As the wall attraction $h$ increases, the interface undergoes a transition from a delocalized phase, in which contacts with the wall have vanishing density, to a localized phase with a positive density of contacts. For sufficiently large $\beta$, the homogeneous wetting point and its sharp near-critical free-energy behavior are known. We prove that, for every fixed $\alpha\ge0$, the quenched critical point coincides with the homogeneous wetting point, $h_c(\beta,\alpha)=h_w(\beta)=-\log(1-e^{-4\beta})$. Moreover, writing $u=h-h_w(\beta)$, we show that the quenched excess free energy satisfies $\overline F_\beta(\alpha,u)=\overline F_{\mathrm{homo}}(\beta,u)+o(u^3)$ as $u\downarrow0$, where $\overline F_{\mathrm{homo}}(\beta,u)$ denotes the leading homogeneous wetting asymptotic and satisfies $\overline F_{\mathrm{homo}}(\beta,u)\asymp u^3$. Thus, at the level of both the critical point and the leading near-critical free-energy asymptotics, the quenched model has the same behavior as the homogeneous SOS wetting model. This contrasts with the disordered SOS pinning model, in which quenched disorder leaves the critical point unchanged but modifies the leading critical behavior of the free energy.

We prove that every event invariant under swaps has probability zero or one for either the free or the wired spanning forest on any connected, locally finite, infinite network. Applications include the constancy of the number of components and of the total number of excessive ends. For the free forest, these results resolve longstanding questions of Benjamini, Lyons, Peres, and Schramm [Ann. Probab. (2001), Question 15.7] on constancy of the component count and of Lyons, Morris, and Schramm [Electron. J. Probab. (2008), Question 7.7] on the zero-one law for one-endedness. In the latter case, we obtain the stronger conclusion that the total number of excessive ends is deterministic.

Let $\delta^*_{k,r}$ be the fractional $K_r^{(k)}$-decomposition threshold in minimum codegree. For fixed $k\ge2$, $r\ge k+1$ and $\varepsilon>0$, we prove that every $n$-vertex $k$-uniform hypergraph $G$ with minimum codegree at least $(\delta^*_{k,r}+\varepsilon)n$ admits, with high probability, a fractional $K_r^{(k)}$-decomposition after retaining each edge independently with probability $p \ge C(\log n/n^{r-k})^{1/(\binom{r}{k}-1)}$. The bound on $p$ is sharp up to a constant factor, confirming the conjectured threshold order in the graph case. The proof develops a probabilistic matrix-scaling approach.

Understanding how macroscopic wealth distributions emerge from microscopic transaction rules among agents is a central problem in econophysics. While many traditional models restrict agents' wealth to non-negative values, debt is an essential feature of realistic economic systems. In this paper, we introduce a stochastic transaction mechanism into the agent-based model with a central-bank-mediated collective debt limit, as studied in [14]. In our model, each agent probabilistically determines whether to transfer a coin, depending on their current asset or debt level. From a probabilistic perspective, the resulting system can be formulated as a zero-range process with $\mathbb{Z}$-valued site occupation numbers, where negative values represent debt. We prove that, under an appropriate scaling, the wealth distribution in the stationary state converges to either a two-sided Gamma distribution or a Gamma distribution, depending on the asymptotic behavior of the function governing coin transfers. Our results rigorously identify these previously unconsidered limiting distributions and provide a mathematical framework for understanding how stochastic monetary exchanges give rise to macroscopic wealth distributions.

Catastrophe bonds have become a central instrument for transferring catastrophic risk to capital markets, yet their payout structures are typically specified exogenously rather than optimized. In this paper, we treat the indemnity function as the decision variable and study the optimal design of CAT bonds from the sponsor's perspective. We establish the optimality of a piecewise linear indemnity function under both expected-disutility and spectral-risk-measure frameworks and thereby reduce an infinite-dimensional design problem to a one-dimensional optimization over the attachment point. The optimal design also eliminates the potential moral hazard associated with the discontinuity of threshold payout structures. We characterize the optimal attachment point under six risk criteria and obtain explicit or semi-explicit solutions. We further derive bounds for the optimal attachment point and develop comparative statics that provide practical design insights. Using U.S. severe storm losses from NOAA's Billion-Dollar Weather and Climate Disasters database for calibration, we show that the optimal CAT bond designs achieve substantial risk reductions across most risk criteria.

We prove that the graph of a standard one-dimensional Brownian motion, $\Gamma_B=\{(t,B_t):t \in [0,1]\}$, is almost surely not conformally removable. This resolves the question left open by Doherty and Miller in their study of the Sobolev removability of the Brownian graph~\cite{DM26}. We also construct non-removable $\alpha$-Hölder graphs for every $\alpha \in (0,2/3)$, including the previously open range $\alpha \in [1/2, 2/3)$. Together with the known removability result for $\alpha>2/3$, this identifies $2/3$ as the threshold for conformal removability of Hölder graphs.

Let $(\xi_t)_{t \geq 0}$ be a continuous-time Markov process taking values in a countable state space $E$. We consider $k$ copies of $\xi$ started from distinct positions, and write $X = (X_t)_{t \geq 0}$ for the resulting process of configurations in $E^k$. We compare two forms of interaction between the particles: the first is conditioning the particles to never collide (noncolliding dynamics), and the second is simply switching off any jumps that would cause two particles to occupy the same position (exclusion dynamics). We show under mild conditions on the tail probabilities of the collision time that the dynamics are related by an explicit exponential martingale change of measure of one another. This generalises a recent result in the integrable probability literature relating TASEP on the ring to asymmetric walkers conditioned to never collide.

We study total path length in recursive trees with positive deterministic attachment weights. Writing $W_n=\sum_{i=1}^n w_i$ and $p_n=w_n/W_n$, we obtain exact martingale-innovation identities and a variance recurrence. Under the condition $p_n=O(n^{-1})$, centered total path length divided by $n$ converges almost surely and in $L^2$ to a nondegenerate random variable, and its variance is asymptotic to a positive constant times $n^2$. No polynomial asymptotic for $W_n$ is required. For power weights $w_i=i^\alpha$, the same argument applies to every real $\alpha$, including the critical and summable regimes beyond the positive-power cumulative-weight assumptions of existing profile theory. The expected average depth is logarithmic for $\alpha>-1$, iterated logarithmic for $\alpha=-1$, and bounded for $\alpha<-1$, while the global fluctuation scale remains linear throughout. In the summable regime we identify the random limit through the weighted depths of the infinite tree. The uniform case recovers the classical variance coefficient $2-\pi^2/6$. For linear weights we evaluate the coefficient as $8-2\pi^2/3$. Although this tree and a random binary search tree have identical insertion-depth marginals and expected total path length, their asymptotic variance coefficients differ by one. This gives an explicit comparison of global dependence that is invisible in individual depth distributions.

In this paper, we give exact combinatorial expressions for all CUE joint moments on the unit circle, in the case that all exponents are even positive integers, yielding the corresponding asymptotics. We offer two equivalent expressions for these quantities: the first is a sum involving Kostka numbers; the second is related to counting integer matrices (magic squares). As an application of our results, we unify many results in the literature, which were previously disconnected. Moreover, we prove a conjecture of Basor et al. (2018) regarding an irregular conformal block structure.

We study negative edge correlation for two well-known models in statistical physics, the arboreal gas and the $q < 1$ random-cluster model. We show that after the edges $e$ and $f$ are removed, their Rayleigh difference is a crossing contribution minus the covariance of two endpoint-connectivity events. For the arboreal gas on an arbitrary finite graph, we analyze the two leading coefficients of this difference. The first is the classical transfer-current square. If it vanishes, the second has an electrical sum-of-squares formula. This proves negative edge correlation at sufficiently large fugacity whenever one of the two leading coefficients of the Rayleigh difference is nonzero, and characterizes simultaneous equality by an equipotential decomposition. For the random-cluster model on the complete graph $K_n$, we prove negative edge correlation throughout $q<1$ when all edge weights are at least $2$. In the uniform case, we prove something stronger:negative correlation holds when all the weights are equal and at least $1$. In particular, distinct edges in a connected spanning subgraph of $K_n$, weighted by a fixed fugacity, are negatively correlated for every $n$. We also explore what seems to be a general phenomenon: positive correlation of connectivity events. For the arboreal gas, we prove positive correlation for connectivity events at high fugacities. On lattices, the conjectured inequality would make the two-point function supermultiplicative and produce a convex inverse correlation length. Exact computations support the graph and matroid conjectures, including the Seymour–Welsh matroid $\mathcal S_8$, where ordinary edge-negative correlation fails. The results in this paper were derived by the authors without the use of Large Language models. The authors did benefit from using GPT-5 Pro for generating code to test out hypotheses as well as for simplifying the arguments.

We study a class of Cauchy problems driven by pseudo-differential operators with Lévy-type symbols. We extend earlier results from exponential initial data to a broader set of initial conditions. We demonstrate the existence and uniqueness of the solution, which has an explicit convolution representation with the associated kernel. We then derive an asymptotic expansion of the solution, with coefficients expressed in terms of moments and, after taking logarithms, in terms of cumulants. These expansions are organized through generalized Feynman graph representations, providing a clear combinatorial description of both the solution and its logarithm. We also establish a Borel summability result under suitable conditions and recover the classical exponential case as a special instance. Finally, we illustrate the convolution kernel and solution through a one-dimensional Lévy-Gaussian example.

We establish non-asymptotic risk bounds for power Fréchet means in geodesic metric spaces with curvature bounded above. The observations form weighted, possibly infinite sequences of independent random variables whose laws and means may differ. We treat three settings. For $2$-Fréchet means in Hadamard spaces, we obtain a sharp mean squared error bound that becomes an identity in Hilbert spaces. For $2$-Fréchet means in CAT($\kappa$) spaces with $\kappa>0$, we establish variance and Wasserstein contraction inequalities with optimal constants depending on the circumradius $\varrho$ of the closed convex data domain, and obtain mean squared error bounds throughout the maximal range $\varrho<\pi/(2\sqrt{\kappa})$. For $\alpha$-Fréchet means in Hadamard spaces, $1<\alpha<2$, we derive finite $L^\alpha$ risk bounds under weighted $\alpha$-moment conditions, allowing even an infinite $\alpha$-moment of the population mixture. The proofs combine variance, quadruple, and contraction inequalities with a leave-one-out stability technique. Applications give prior and posterior bounds for Dirichlet-process Fréchet means and finite-sample guarantees for local constant Fréchet regression. The regression results require no response-space entropy condition and replace density smoothness assumptions common in earlier work with transport smoothness; for $\alpha<2$, the bounds remain finite even when the responses have infinite variance.

The rough response calculus for the rotationally marked SLE welding law gives a closed Weil–Petersson energy but does not by itself give a process on rough weldings. For $0<\kappa\leq4$, we prove that this form is quasi-regular in the natural topology of circle homeomorphisms and therefore generates a canonical diffusion reversible with respect to the SLE welding probability. The proof constructs a compact form nest from inverse arc masses. We identify the generator as $-D^*D$ and prove absolute $L^2$ convergence of its Fourier expansion on the response cylinder core, together with the Fukushima decomposition. A common Fourier noise and planar estimates preventing boundary collapse yield a second quasi-regular realization by normalized Jordan welding triples. The diffusion is irreducible and converges strongly in $L^2$ to equilibrium. For $0<\kappa<4$, the complement of the conformally removable welding locus has zero capacity.

We prove that every $n$-point set in the plane has a two-coloring with discrepancy $O(n^{1/4})$ for all triangles. This improves the previous bound $O(n^{1/4}\sqrt{\log n})$ and matches the classical lower bound.

Let $\Gamma<\mathrm{Isom}(\mathbb H^n)$, $n\geq2$, be a cocompact lattice containing a convex cocompact codimension-one subgroup in the sense of Sageev. We prove that every finitely supported admissible random walk on $\Gamma$ has hitting measure singular with respect to Lebesgue measure on $\partial\mathbb H^n$. The proof extends the method of Kosenko–Tiozzo, replacing Fourier analysis for a cyclic subgroup by a von Neumann dimension argument that also applies to nonabelian subgroups. As corollaries, we prove the singularity conjecture of Kaimanovich and Le Prince for cocompact hyperbolic lattices admitting a proper cocompact cubulation, cocompact Kleinian groups, and cocompact hyperbolic lattices with totally geodesic sublattices of codimension one. In particular, our result covers all cocompact hyperbolic reflection groups, cocompact arithmetic lattices of simplest type and cocompact nonarithmetic lattices arising from the hybrid and inbreeding constructions.