math.PR

We study second-order Consensus-Based Optimization (CBO), a derivative-free global optimization algorithm in which the consensus force and the multiplicative exploratory noise act on particle velocities. We prove quantitative uniform-in-time propagation of chaos for the unmodified second-order CBO dynamics, together with an almost uniform-in-time stability estimate for the microscopic particle system. The proof is not a direct adaptation of the first-order CBO argument. Although both first- and second-order CBO have multiplicative noise that degenerates near consensus and a shift-invariant weighted interaction, the kinetic model has an additional structural obstruction: the consensus mechanism and the stochastic forcing act only on the velocity variable, while the position variable evolves by transport. Thus spatial concentration has to be recovered indirectly through velocity dissipation. Moreover, the shift-invariant interaction leaves a translation mode that is not directly damped by the consensus force, so a standard synchronous coupling in the Euclidean phase-space distance does not close uniformly in time. The main idea of the paper is to introduce shifted internal variables that separate the contracting fluctuation modes from the undamped translation mode. In these variables we build a Lyapunov functional with a position-velocity cross term and prove exponential decay of centered moments. This decay is the mechanism that makes the time-dependent coupling coefficient integrable. Combining it with uniform-in-time raw moment bounds, concentration inequalities, stability estimates for the weighted mean, and a Monte Carlo estimate, we obtain the classical Monte Carlo rate for propagation of chaos uniformly in time. The system-to-system stability estimate avoids the sampling error and yields the faster rate \(O(J^{-q})\).

We compute the exact log-asymptotics of the persistence probability, and determine the entropic repulsion profile conditioned on persistence, for general $d$-dimensional stationary Gaussian fields with spectral singularity at the origin of order $\alpha \in [0,d)$. Under mild regularity conditions these are shown to be universal, depending only on $\alpha$ and $d$, and to have explicit formulations in terms of the capacity and equilibrium potential of the $\alpha$-Riesz kernel. This generalises a result of Bolthausen, Deuschel and Zeitouni on the Gaussian free field to a wide class of Gaussian fields with spectral singularity.

We study the local geometry of the three-dimensional uniform spanning tree and its connection with the Abelian sandpile model. We obtain sharp tail exponents, up to subpolynomial errors, for the past of the origin in the three-dimensional UST and for the $0$-tree of the $0$-wired uniform spanning forest. As a principal application, we prove the corresponding three-dimensional Abelian sandpile avalanche exponents: the avalanche-cluster radius has tail exponent $1$, while both the avalanche-cluster size and the total number of topplings have tail exponent $1/3$. These results identify the leading power-law behaviour of three-dimensional sandpile avalanches and improve previously known bounds.

We study nonconcentration of hitting times for simple random walk on finite graphs. We prove that, for every connected graph with $n$ vertices, \[ \operatorname{Var}_x(\tau_y)+\mathbb E_x\tau_y \ge \frac{(\mathbb E_x\tau_y)^2}{1+\log n}, \] with the logarithmic term sharp up to constants. Under a bounded-degree assumption the additive mean term can be removed, giving a variance lower bound depending only on \(\mathbb E_x\tau_y\) and the graph distance \(\dist(x,y)\). We show that this degree assumption is necessary by constructing high-degree graphs with linear mean and bounded variance; the same construction disproves a conjecture of Norris-Peres-Zhai concerning local nonconcentration of hitting times. We also prove a sharper tree estimate, extend the main argument to finite reversible Markov chains, and show that Holroyd's interval conjecture, stated in Norris-Peres-Zhai, fails even for bounded-degree trees.

We classify measures on $\{0,1\}^{\mathbb{Z}^d}$, $d \geq 3$, the space of subsets of $\mathbb{Z}^d$, which are invariant under all affine special linear transformations. In other words, we classify simple point processes on $\mathbb{Z}^d$ whose law is invariant under affine special linear transformations. We show that every such process is built from a random equivariant polynomial together with independent random sampling, a higher-order generalisation of the cut-and-project method: a random polynomial map is drawn from a distribution invariant under a natural action of $\mathrm{SL}_d(\mathbb{Z})$, each site is then retained independently with a probability determined by a measurable function of the polynomial's value, and the classical cut-and-project construction is recovered in the degree-one case. As a corollary, when the underlying $\mathbb{Z}^d$-action is weakly mixing the measure must be a convex combination of Bernoulli shifts, in the spirit of de Finetti's theorem on exchangeable processes. Our theorem also makes precise how the Howe--Moore theorem fails for the pair $(\mathrm{ASL}_d(\mathbb{Z}), \mathrm{SL}_d(\mathbb{Z}))$. Motivated by this classification, we formulate a conjecture for $\mathrm{ASL}_d(\mathbb{R})$-invariant point processes on $\mathbb{R}^d$, predicting that any such set decomposes into a Poisson part and a quasicrystal part. The proofs rely on the interaction between the Host--Kra theory of characteristic factors, Zimmer's theory of dynamical cocycles of simple Lie groups, and the dynamics of $\mathrm{SL}_d(\mathbb{Z})$-actions on homogeneous spaces.

We provide an explicit probability measure on $\mathbb{R}$ for which the fifth time derivative of the entropy along the heat flow is positive at some time. This disproves the Gaussian completely monotone (GCM) conjecture (Cheng-Geng '15) and therefore also the Gaussian optimality conjecture (McKean '66) and the entropy power conjecture (Toscani '15). Our proof also implies the existence of a log-concave probability measure on $\mathbb{R}$ for which the GCM conjecture fails at some order. The explicit counterexample was found by GPT-5.5 Pro.

We study an online vector balancing problem, in which $n$ independent Gaussian random vectors $\boldsymbol{\zeta}(1),\dots,\boldsymbol{\zeta}(n) \sim \mathcal{N}(0, I_n)$, each of dimension $n$, arrive one at a time. The goal is to choose signs $\varepsilon(1),\dots,\varepsilon(n) \in \{\pm 1\}$ with $\varepsilon(k)$ depending only on $\boldsymbol{\zeta}(1),\dots,\boldsymbol{\zeta}(k)$, so as to minimize the expected $\ell^{\infty}$ norm of the signed sum $\frac{1}{\sqrt{n}}\sum_{k = 1}^n \varepsilon(k) \boldsymbol{\zeta}(k)$. Prior work showed that the optimal value $V^n$ is $O(1)$, at least for Rademacher $\boldsymbol{\zeta}(k)$'s, by constructing specific algorithms. Our main contribution is to determine the exact limit $V^{\infty} = \lim_{n\to\infty} V^n$ as the value of a nonstandard stochastic control problem of mean-field type: find the narrowest terminal interval into which a Brownian motion can be adaptively steered under a uniform-in-time $L^2$ constraint on the drift. The proof of the lower bound $V^{\infty} \leq \liminf_{n \to \infty} V^n$ uses probabilistic compactness arguments, and is very flexible. In fact, we show that the lower bound is universal, in that it holds as long as the entries of the $\boldsymbol{\zeta}(k)$ vectors are i.i.d. with mean zero, variance 1, and finite fourth moment. The proof of the upper bound $\limsup_{n \to \infty} V^n \leq V^{\infty}$ is more delicate, relying on dynamic programming principles and a priori bounds obtained from a coupling procedure involving the F\"ollmer drift, which makes explicit use of the Gaussian structure. In addition to our main convergence result, we provide some analysis and asymptotics for the limiting mean-field control problem.

It is a fundamental question in epidemiology to estimate, model and predict the growth rate of a pandemic. Analogously, analysing the diffusion of innovation, (fake) news, memes, and rumours is of key importance in the social sciences. The resulting epidemic growth curves can be classified according to their growth rates. These have been found to range from exponential to both faster super-exponential curves and slower subexponential or polynomial curves. Previous research has lacked a unified explanatory framework capable of accommodating super-exponential, (stretched) exponential, and polynomial growth patterns within the same contact network. In this paper we propose a simple agent-based network model that can capture all these phases. We provide such a framework by modelling how transmission rates depend on spatial distance and on individuals' numbers of contacts. By comparing the growth rate of spreading processes with or without degree-dependent and/or distance-dependent contact rates through data-driven and synthetic simulations on real and modelled networks with underlying geometry, we find evidence that even a 'sublinear presence' of these causes may cause a significant slow down of the growth rate on the same underlying network. We find that the growth rate is governed by a combination of three factors: geometry, the prevalence of weak ties, and superspreaders. We confirm our results with rigorous proofs in a theoretical model, using a spatial multiscale-argument in long-range heterogeneous first passage percolation. Our results give a plausible explanation of why the consecutive waves of a single pandemic can differ in their growth even if their spreading mechanisms are similar.