ALT We introduce a quadratic ancestor inequality for weighted spanning trees on finite undirected graphs and give two applications. First, for every finite n-vertex transitive graph, the expected distance in its uniform spanning tree (UST) between two independent uniform vertices is at least √n-1, resolving a conjecture in [Benjamini–Kozma, Comm. Math. Phys. 259 (2005), 257–286]. Second, we prove that the growth exponent of three-dimensional loop-erased random walk (LERW) is at least 3/2.

ALT Compactified Imaginary Liouville Theory (CILT for short) is a probabilistically constructed Conformal Field Theory (CFT), expected to have connections to minimal models, Coulomb gas integrals, and loop models. In this paper, we study the quantization of Segal's semigroup given by CILT. Each element 𝐓f in the quantized semigroup corresponds to a conformal annulus: the unit disk D with f(D°) removed, where f:D→ f(D)⊂ D° is a contracting biholomorphism. This semigroup encodes the local conformal structure of the CFT, in particular the Virasoro symmetry with central charge <1 for CILT. We first relate the quantized Segal's semigroup to Segal's amplitudes of annuli and recover the Virasoro generators 𝐋ₙ,𝐋̃ₙ (n≥0) of CILT as generators of this semigroup. Compared to the real Liouville CFT, the main difficulty here is the absence of positivity, which we address through estimates for the imaginary Gaussian Multiplicative Chaos and non-self-adjoint operators. Using Segal's gluing axiom as in

ALT We characterize the cases in which the minimal eigenvalues of interchange-process Laplacians on two Specht modules can be compared up to a positive multiplicative constant, independent of the edge rates. In the comparable cases, we establish an inverse-polynomial lower bound for this constant. We conjecture that the constant can always be taken to be 2/3. We apply these ideas to the mixing times of the interchange and exclusion processes. Using the comparison method of Hermon, Kozma and Salez, we prove that our conjecture implies a conjecture of Oliveira on the mixing time of the interchange process. We also obtain unconditional mixing-time bounds for both processes.

ALT Consider the diffusions dXₜε=b(Xₜε)dt+√(2ε)σ(Xₜε)dBₜ on ℝᵈ, where b and σ are smooth coefficients and Bₜ is an r-dimensional Brownian motion, with r≤ d. We allow nonreversible dynamics and a degenerate diffusion matrix. Let ℐε be the level-two large deviation rate functional of the empirical measure. Under our assumptions, including a complete metastable hierarchy described by resolvent convergence, we establish the full Γ-expansion ℐε=1/εJ⁽⁻¹⁾+J⁽⁰⁾+∑ₚ₌₁𝔮1/θε⁽ᵖ⁾J⁽ᵖ⁾ in the sense of successive Γ-limits in the weak topology on probability measures. Here θε⁽ᵖ⁾ are the metastable time scales, and each J⁽ᵖ⁾, p≥1, is the lift of the level-two rate functional of the corresponding finite-state Markov chain. We establish the pre-metastable Γ-limits for generalized overdamped, underdamped, and third-order Langevin dynamics. For the generalized overdamped model, existing metastability results yield the full expansion under additional conditions. For the underdamped and third-order models, the fu

ALT We study stopping sets and continuous-time decision trees (CTDTs) for Poisson point processes. We first give a proof of the spatial Markov property without assuming that the stopping set contains finitely many points or admits a finite approximation. The proof uses the multivariate Mecke formula on the complement of the stopping set. We then prove that every μ-continuous CTDT has an explicit exploration representation: between successive discoveries, the explored set grows according to a deterministic rule depending on the points already observed. If μ is diffuse, the points are revealed one by one almost surely; this need not be imposed as a separate assumption. We also allow atoms of μ, in which case all points at one location are revealed together. Finally, for diffuse μ, we prove that a locally bounded stopping set revealing at most one point is attainable by a μ-continuous exploration. More generally, an increasing sequence of locally bounded stopping sets, each revealing at most

ALT We study asymptotic expansions for nonlinear filtering in the small-system-noise regime. Our main result is that the expansion coefficients of conditional cumulants satisfy a closed finite-dimensional system that can be solved recursively. At each expansion degree, the evolution of a coefficient introduces a coefficient of the next higher cumulant rank, while a high-rank vanishing property terminates this hierarchy, yielding a finite triangular tower. The first layer is the Kalman–Bucy filter, and subsequent layers satisfy linear equations once the lower layers are known. To derive these equations, we establish generalized Zakai and Kushner–Stratonovich equations for observation-dependent random variables and obtain explicit coefficient equations in arbitrary dimension. The conditional cumulant expansion also determines the expansion of the conditional characteristic function. We further obtain an explicit bound on the number of scalar variables in terms of the state dimension and expa

ALT We consider a randomly twisted hypercube on N vertices and study a simple or lazy random walk on this graph. We show that for both walks the mixing time is of order log N, and the model does not exhibit cutoff. We also establish that both walks have cover time of order Nlog N, and we show that the chromatic number of the graph diverges.

ALT Let X₁,X₂,… be i.i.d. random variables with zero mean and variance σ²∈(0,∞). In this paper, we establish a sharp Cramér-type moderate deviation for the self-normalized maximum of partial sums under a finite p-th absolute moment, 2<p≤3. In particular, we show that mathbbP(M_n≥ xV_n)=2(1-Φ(x)){1+O(1)β_n,p(1+x)^p} for 0≤ x≤β_n,p^-1/p, where M_n=max₁≤k≤n\sum_j=1^kX_j, V_n=(\sum_j=1^nX_j²)¹/2, β_n,p=E|X₁|^p/(nᵖ⁻²/2σ^p), and Φ is the standard normal distribution function. This proves a conjecture of Shao2010. More precisely, we bound the relative error by the same truncated-moment quantity as for self-normalized sums. The proof compares the maximum with the absolute sum while preserving their common denominator, through a reflected-walk representation and Fourier estimates with distribution-adapted smoothing.

ALT We investigate the Glauber–Misanthrope process, obtained by superposing a Misanthrope dynamics with finite site capacity and a Glauber dynamics which creates or annihilates particles one at a time at each site. In a perturbative regime around the diffusively scaled partial exclusion process, we prove the cutoff phenomenon and identify the limiting mixing time. Our approach relies on the information percolation framework developed by Lubetzky and Sly and extended by Tran, which we adapt to accommodate multiple occupancy per site and account for the two distinct correlation mechanisms generated by the Misanthrope and Glauber dynamics. Our lower bound on the mixing time holds under assumptions more general than those required for the matching upper bound.

ALT The monkey walk is a model of reinforced random walk introduced by Boyer and Solis-Salas as a model for the exploratory behaviour of capuchin monkeys. At each step of the walk, the walker flips a biased coin to determine whether to take a simple, symmetric random walk step on ℤᵈ or to "relocate", i.e. choose a time uniformly at random in its past and jump to where it was at that time. More recently, Boyer, Falcón-Cortés, Giuggioli and Majumdar looked at a modification of the model, where the probability to relocate is ξ₀ when the walker sits at the origin and ξ>ξ₀ anywhere else. We prove that, for all ξ₀∈ (0,1), if ξ is sufficiently close to 1, then the monkey walk "localises", i.e. the position of the walker at time n, Wₙ, converges in distribution without any renormalisation. (We in fact prove convergence of the occupation measure of the walker.) This is in contrast with the case when ξ=ξ₀, for which it is known that Wₙ/√(log n) converges in distribution to a standard Gaussian (we sa

ALT We consider variants of the simple symmetric exclusion process (SSEP) where particles are 'large'; while in the SSEP, each lattice point is either occupied or not occupied by a single particle, in our model each particle is represented by a fixed shape, and the dynamics is so that shapes cannot overlap. In one dimension, this reduces to a finite range interaction that prevents particles from coming closer to each other than a certain distance. In this case, we prove the existence of the hydrodynamic limit, and identify the relevant macroscopic PDE, which is a porous medium equation. In two dimensions, we have no rigorous results, but instead rely on numerical simulations to approximate the diffusion matrix. Under the assumption that the hydrodynamic limit exists in two dimensions, we derive properties of the diffusion matrix based on the shape of the particles.

ALT We investigate the well-posedness of a backward stochastic porous media equation whose nonlinear term is a polynomial operator corresponding to the slow diffusion regime. The main challenge stems from the structure of this nonlinear operator, which causes the operator to lose maximal monotonicity and prevents the use of standard weak-strong convergence techniques to identify the limit. Moreover, since the Laplace operator makes it difficult to separate the operators and discuss them individually, we attempt to obtain the of limit by investigating the global properties of the operator.

ALT We study branching Brownian motion in a resource environment created by the particles themselves. Particles reproduce at rate β times the resource nearby, resource is consumed at rate c times the number of particles nearby, and each death returns resource mass m. Let θ=βm/c, which represents the number of births supported by resource from one corpse, assuming perfect efficiency. We prove that the population dies out almost surely if θ≤1, even with perfect recycling (θ=1). If θ>1 and the interaction kernel has finite range and is bounded below at short distances, the population survives with positive probability from every initial state with at least two particles, or with one particle and some resource. The threshold does not depend on the diffusivity, the death rate or the interaction kernel. We prove extinction for θ≤ 1 by showing the total mass in the system to be a supermartingale. The proof of survival rests on two estimates: a large colony grows if it consumes its corpses with li

ALT We consider exponential last passage percolation in half-space with a deterministic first row. This model corresponds to a version of TASEP in half-space in which particles enter at the origin and are restricted by a moving wall. We determine the possible limiting distributions of rescaled last passage times depending on the boundary strength and the choice of the wall. As key inputs to our analysis, we establish several results for point-to-point half-space last passage times, including upper tail bounds, weak convergence, and bounds on the probabilities that geodesics touch the diagonal for certain start and end points.

ALT We study small-time asymptotics for kinetic stochastic differential equations with singular drifts in kinetic Kato classes. We establish a small-time large deviation principle and prove a pointwise Varadhan-type asymptotic for the associated transition density.

ALT We study dissipation enhancement for stochastic p(x)-Laplace equations with divergence-free transport noise on Tᵈ, d≥2. Under log-Hölder continuity and degenerate subcritical growth, we establish global variational well-posedness and an exact energy identity. For every prescribed positive coefficient, suitable transport noise yields averaged algebraic decay uniformly on bounded zero-mean L² balls. The noise is independent of the individual initial datum. This uniform enhancement holds despite the absence of homogeneous scaling and the two flux growth regimes associated with the variable exponent. We also identify p⁺ as the optimal exponent for uniform modular coercivity near zero and obtain a strict averaged comparison with the unperturbed evolution at any prescribed time T≥1 on initial-data shells with bounded gradient modular. For each prescribed decay rate and finite moment order, deterministic stepwise Fourier profiles with fixed intensity and increasing frequency range give expone

ALT Mansour, Rastegar and Roitershtein studied the roots of ballistic deposition on a cyclic strip of K sites, namely the sites that receive a particle at height one, and the number Di,K of pairs of consecutive roots at distance i+1. They computed the first two moments of Di,K for i≤7. They conjectured that for every i the mean and the variance are exactly linear in K, that Di,K satisfies a central limit theorem, and that the standardized vector (Di,K)ᵢ≥₁ converges to a Gaussian process. We prove these conjectures. The key observation is that the roots are exactly the valleys of a uniformly random circular arrangement, so that Di,K counts occurrences of a family of consecutive patterns. We determine sizes from which the moments are exactly linear, which cannot be lowered in any case we checked, and give closed forms for the mean and variance constants and for the limiting covariance. We also prove a Wasserstein bound of order (E Di,K)⁻¹/² that is uniform in i, and a local limit theorem. Fi

ALT For the empirical measure of the first n points of one infinite sequence of independent uniform samples in the unit square, we identify the leading constant in the bounded-Lipschitz distance to area measure, equivalently in W₁, both in expectation and almost surely along all sample sizes. The constant is c*=κ₁/(2√π)∈(0,1/4], where κ₁ is the linear-growth Gaussian variational constant of Armegioiu–Goldman–Grotto–Trevisan. This answers the two-dimensional bounded-Lipschitz convergence question posed by Yukich in 1989. We give the complete endpoint argument, square–torus comparison, empirical reduction, and full-sequence probability argument.

ALT The grass-bushes-trees process is a two-type contact process in which the types face asymmetric birth mechanisms. Vertices of a graph can either be occupied by a tree, occupied by a bush, or empty (containing grass). The trees die with rate 1 and give birth with rate λ_T on any neighbouring vertex (possibly taking the place of a bush). Bushes die with rate 1 and give birth with rate λℬ on any empty neighbouring vertex; in particular, they are forbidden from taking the place of trees. We study this process on a power law random graph, namely the erased configuration model. We study which graph parameters lead to the possibility of coexistence versus the necessity of competitive displacement, i.e. joint metastability or fast extinction of the bushes.

ALT We introduce the class of qₓ-Bass martingales, extending the notion of q-Bass martingales. These are martingale couplings with prescribed marginals whose transition kernels are closest to a given family of reference kernels (qₓ)ₓ∈ℝᵈ in the sense of a suitable weak optimal transport problem. We show that qₓ-Bass martingales are optimizers of this problem and establish a strong duality result. Building on this duality, we introduce an algorithm for their computation and, under compactness assumptions on the prescribed marginals, prove its convergence, thereby establishing existence. We then specialize to one-dimensional marginals and show that the algorithm reduces to a fixed-point iteration.