We investigate the relationship between curvature bounds and local functional inequalities. In particular, we show that local inequalities derived through the maximum principle approach of Ivanisvili and Volberg (In 50 Years with Hardy Spaces (2018) 281–305, Birkhäuser/Springer) are equivalent to lower bounds on the Ricci curvature in the sense of Bakry–Emery. Moreover, we will develop this Euclidean space technique to metric measure spaces satisfying the RCD condition, providing a unified approach to functional and isoperimetric inequalities in nonsmooth spaces with a synthetic Ricci curvature bound. Finally, we are interested in commutation properties for semigroup operators on Rn in the absence of positive curvature, based on a local eigenvalue criteria.
We study the parabolic Anderson model (PAM) {(partial derivative)/ (partial derivative t )u(t,x)=(1)/(2 )Delta u(t,x)+u(t,x)xi(x) t >0, x is an element of R d, u(0,x)equivalent to 1 x is an element of R (d), where xi is spatial white noise on R-d with d is an element of {2,3}. We show that the peaksof the PAM are macroscopically multifractal. More precisely, we prove thatthe spatial peaks of the PAM have infinitely many distinct values, and wecompute the macroscopic Hausdorff dimension (introduced by Barlow andTaylor (J. Phys. A: Math. Gen.22(1989) 2621-2628;Proc. Lond. Math.Soc. (3)64(1992) 125-152)) of those peaks. As a byproduct, we obtain theexact spatial asymptotics of the solution of the PAM at any fixed, sufficientlylarge time. We also study the spatiotemporal peaks of the PAM and showtheir macroscopic multifractality. Some of the major tools used in our prooftechniques include paracontrolled calculus and tail probabilities of the largestpoint in the spectrum of theAnderson Hamiltonian.
In this paper, we present a unified approach to establish the uniqueness of generalized conformal restriction measures with central charge c ∈ (0, 1] in both chordal and radial cases, by relating these measures to the Brownian loop soup. Our method also applies to the uniqueness of the Malliavin-Kontsevich-Suhov loop measures for c ∈ (0,1], which was recently obtained in [Baverez-Jego, arXiv:2407.09080] for all c ≤ 1 from a CFT framework of SLE loop measures. In contrast, though only valid for c ∈ (0,1], our approach provides additional probabilistic insights, as it directly links natural quantities of MKS measures to loop-soup observables.
We prove the global well-posedness of the dynamical sine-Gordon model up to the third threshold, that is, for parameters beta(2) < 6 pi. The key novelty in our approach is the introduction of the so-called resonant equation, whose solution is entirely deterministic and completely captures the size of the solution to the dynamical sine-Gordon model. The probabilistic fluctuations in the dynamical sine-Gordon model are then controlled using uniform estimates for modified stochastic objects.
In this paper we construct a stochastic process, more precisely, a (nonlinear) Markov process, which is related to the parabolic p-Laplace equation in the same way as Brownian motion is to the classical heat equation given by the (2-) Laplacian.
We prove a large deviations principle for the probabilistic Schwarzian Field Theory at low temperatures. We demonstrate that the good rate function is equal to the action of the Schwarzian Field Theory, and we find its minimisers. In addition, we define an analogue of the H & ouml;lder condition on the functional space Diff1(ZC)/PSL(2, IIg) in terms of cross-ratio observables, characterise them in terms of the usual H & ouml;lder property on the space of continuous functions, and deduce the corresponding compact embedding theorem. We also show that the Schwarzian measure concentrates on functions satisfying the defined condition.
For (λ_1,...,λ_d)=λ∈(0,1)^d with λ_1>...>λ_d, denote by μ_λ the Bernoulli convolution associated to λ. That is, μ_λ is the distribution of the random vector ∑_n≥0±(λ_1^n,...,λ_d^n), where the ± signs are chosen independently and with equal weight. Assuming for each 1≤ j≤ d that λ_j is not a root of a polynomial with coefficients ±1,0, we prove that the dimension of μ_λ equals min{_Lμ_λ,d}, where _Lμ_λ is the Lyapunov dimension. More generally, we obtain this result in the context of homogeneous diagonal self-affine systems on ℝ^d with rational translations. The proof extends to higher dimensions the works of Breuillard and Varjú and Varjú regarding Bernoulli convolutions on the real line. The main novelty and contribution of the present work lies in an extension of an entropy increase result, due to Varjú, in which the amount of increase in entropy is given explicitly. The extension of this result to the higher-dimensional non-conformal case requires significant new ideas.
We study the eigenvalue distribution and resolvent of a Kroneckerproduct random matrix model A (R) In & times;n + In & times;n (R) B + Theta (R) Xi is an element of & Copf;n2 & times;n2 , where A, B are independent Wigner matrices and Theta, Xi are deterministic and diagonal. For fixed spectral arguments, we establish a quantitative approximation for the Stieltjes transform by that of an approximating free operator and a diagonal deterministic equivalent approximation for the resolvent. We further obtain sharp estimates in operator norm for the n & times; n resolvent blocks and show that off-diagonal resolvent entries fall on two differing scales of n-1/2 and n-1, depending on their locations in the Kronecker structure. Our study is motivated by consideration of a matrix-valued least-squares optimization problem minX is an element of & Ropf;n & times;n 12 parallel to X A+B X parallel to 2F+ 1 & sum;ij xi i theta j x2 2 ij subject to a linear constraint. For random instances of this problem defined by Wigner inputs A, B, our analyses imply an asymptotic characterization of the minimizer X and its associated minimum objective value as n -> infinity.
The (m, b) Maker-Breaker percolation game on (Z(2))p, introduced by Day and Falgas-Ravry, is played in the following way. Before the game starts, each edge of Z(2) is removed independently with probability 1-p. After that, Maker chooses a vertex v0 to protect. Then in each round Maker and Breaker claim, respectively, m and b unclaimed edges of Z(2). Breaker wins if after the removal of the edges claimed by him the component of v0 becomes finite, and Maker wins if she can indefinitely prevent Breaker from winning. We show that for any p < 1, almost surely Breaker has a winning strategy for the (1, 1) game on (Z(2))p. This fully answers a question of Day and Falgas-Ravry, who showed that for p = 1 Maker has a winning strategy for the (1, 1) game. Further, we show that in the (2, 1) game on (Z(2))p almost surely Maker has a winning strategy whenever p > 0.9402, while almost surely Breaker has a winning strategy whenever p < 0.5278. This shows that the threshold value of p above which Maker has a winning strategy for the (2, 1) game on Z(2) is nontrivial. In fact, we prove similar results in various settings, including other lattices and biases (m, b). These results extend also to the most general case, which we introduce, where each edge is given to Maker with probability alpha and to Breaker with probability beta before the game starts.
We establish the quenched local limit theorem for reversible random walk on Zd (with d >= 2) among stationary ergodic random conductances that permit jumps of arbitrary length. The proof is based on the weak parabolic Harnack inequalities and on-diagonal heat-kernel estimates for long-range random walks on general ergodic environments. In particular, this partly solves (Probab. Theory Related Fields 180 (2021) 847-889), Open Problem 2.7, where the quenched invariance principle was obtained. As a byproduct, we prove the maximal inequality with an extra tail term for long-range reversible random walks, which in turn yields the everywhere sublinear property for the associated corrector.
The Horton-Strahler number – also called the register function – is a combinatorial tool that quantifies the branching complexity of a rooted tree. We study the law of the Horton-Strahler number of stable Galton-Watson trees conditioned to have size n (including the Catalan trees), which are the finite-dimensional marginals of stable Lévy trees. While these random variables are known to grow as a multiple of ln n in probability, their fluctuations are not well understood because they are coupled with deterministic oscillations. To rule out the latter, we introduce a real-valued variant of the Horton-Strahler number. We show that a rescaled exponential of this quantity jointly converges in distribution to a measurable function of the scaling limit of the trees, i.e. the stable Lévy tree. We call this limit the Strahler dilation and we discuss its similarities with the Horton-Strahler number.
Given any closed Riemannian manifold $M$, we construct a reversible diffusion process on the space ${\mathcal P}(M)$ of probability measures on $M$ that is (i) reversible w.r.t.~the entropic measure ${\mathbb P}^\beta$ on ${\mathcal P}(M)$, heuristically given as $$d\mathbb{P}^\beta(\mu)=\frac{1}{Z} e^{-\beta \, \text{Ent}(\mu| m)}\ d\mathbb{P}^*(\mu);$$ (ii) associated with a regular Dirichlet form with carr\'e du champ derived from the Wasserstein gradient in the sense of Otto calculus $${\mathcal E}_W(f)=\liminf_{g\to f}\ \frac12\int_{{\mathcal P}(M)} \big\|\nabla_W g\big\|^2(\mu)\ d{\mathbb P}^\beta(\mu);$$ (iii) non-degenerate, at least in the case of the $n$-sphere and the $n$-torus.
We prove nonasymptotic matrix concentration inequalities for the spectral norm of (sub)gaussian random matrices with centered independent entries that capture fluctuations at the Tracy-Widom scale. This considerably improves previous bounds in this setting due to Bandeira and Van Handel, and establishes the best possible tail behavior for random matrices with an arbitrary variance pattern. These bounds arise from an extremum problem for nonhomogeneous random matrices: among all variance patterns with a given sparsity parameter, the moments of the random matrix are maximized by block-diagonal matrices with i.i.d. entries in each block. As part of the proof, we obtain sharp bounds on large moments of Gaussian Wishart matrices.
We study a spatial Markovian particle system with pairwise coagulation, a spatial version of the Marcus--Lushnikov process: according to a coagulation kernel $K$, particle pairs merge into a single particle, and their masses are united. We introduce a statistical-mechanics approach to the study of this process. We derive an explicit formula for the empirical process of the particle configuration at a given fixed time $T$ in terms of a reference Poisson point process, whose points are trajectories that coagulate into one particle by time $T$. The non-coagulation between any two of them induces an exponential pair-interaction, which turns the description into a many-body system with a Gibbsian pair-interaction. Based on this, we first give a large-deviation principle for the joint distribution of the particle histories (conditioning on an upper bound for particle sizes), in the limit as the number $N$ of initial atoms diverges and the kernel scales as $\frac 1N K$. We characterise the minimiser(s) of the rate function, we give criteria for its uniqueness and prove a law of large numbers (unconditioned). Furthermore, we use the unique minimiser to construct a solution of the Smoluchowski equation and give a criterion for the occurrence of a gelation phase transition.
We study the convergence problem for mean field control, also known as optimal control of McKean-Vlasov dynamics. We assume that the data is smooth but not convex, and thus the limiting value function 𝒰 :[0,T] ×𝒫_2(ℝ^d) →ℝ is Lipschitz, but may not be differentiable. In this setting, the first and last named authors recently identified an open and dense subset 𝒪 of [0,T] ×𝒫_2(ℝ^d) on which 𝒰 is 𝒞^1 and solves the relevant infinite-dimensional Hamilton-Jacobi equation in a classical sense. In the present paper, we use these regularity results, and some non-trivial extensions of them, to derive sharp rates of convergence. In particular, we show that the value functions for the N-particle control problems converge towards 𝒰 with a rate of 1/N, uniformly on subsets of 𝒪 which are compact in the p-Wasserstein space for some p > 2. A similar result is also established at the level of the optimal feedback controls. The rate 1/N is the optimal rate in this setting even if 𝒰 is smooth, while, in general, the optimal global rate of convergence is known to be slower than 1/N. Thus our results show that the rate of convergence is faster inside of 𝒪 than it is outside. As a consequence of the convergence of the optimal feedbacks, we obtain a concentration inequality for optimal trajectories of the N-particle problem started from i.i.d. initial conditions.
We investigate the bond percolation model on transient weighted graphs G induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in G have polynomial volume growth with growth exponent alpha and that the Green's function for the random walk on G exhibits a power law decay with exponent v, in the regime 1 <= v <= alpha/2 . In particular, this includes the cases of G = & Zopf;(3) for which v = 1, and G = & Zopf;(4) for which v = alpha/2 = 2. For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance R, like R-v/2+o(1). Our results are, in fact, more precise and yield logarithmic corrections when v > 1 as well as corrections of order log log R when v = 1. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when v > 1 and essentially optimal when v = 1. This extends previous results from (Invent. Math. 232 (2023) 229-299; Ann. Probab. 48 (2020) 1411-1435).
We derive explicit estimates for the asymptotics of the first-passage function for random walks on free groups supported on the powers of standard generators, and use them to prove the singularity of the hitting measure for a similarly defined class of random walks on Fuchsian groups.
Suppose that B is a one-dimensional Brownian motion and let Γ = { (t, B_t) : t ∈ [0,1]} be the graph of B|_[0,1]. We characterize the Sobolev removability properties of Γ by showing that Γ is almost surely not W^1,p–removable for all p ∈ [1, ∞) but is almost surely W^1,∞–removable.
This paper studies the analytic continuation of Liouville eigenstates and shows that they assemble into irreducible highest-weight representations of the Virasoro algebra, for all values of the conformal weights. This builds on previous results from the first author and Guillarmou, Kupiainen, Rhodes Vargas, where such representations were constructed except for the conformal weight on the Kac table. In order to extend these results to the degenerate weights, we find explicit analytic expressions for the Virasoro descendants and uncover the probabilistic meaning of the Kac table. In the algebraic approach to conformal field theory, the irreducibility is a crucial property that must be satisfied by the representations in the spectrum, and is usually taken as an axiom. Computationally, it leads to the celebrated null-vector (or BPZ) equations for correlation functions and conformal blocks, which are the cornerstone of the integrability of the theory.
We study the convergence properties of Glauber dynamics for the random field Ising model (RFIM) with ferromagnetic interactions on finite domains of $\mathbb{Z}^d$, $d \ge 2$. Of particular interest is the Griffiths phase where correlations decay exponentially fast in expectation over the quenched disorder, but there exist arbitrarily large islands of weak fields where low-temperature behavior is observed. Our results are twofold: 1. Under weak spatial mixing (boundary-to-bulk exponential decay of correlations) in expectation, we show that the dynamics satisfy a weak Poincar\'e inequality -- equivalent to large-set expansion -- implying algebraic relaxation to equilibrium over timescales polynomial in the volume $N$ of the domain, and polynomial time mixing from a warm start. From this we construct a polynomial-time approximate sampling algorithm based on running Glauber dynamics over an increasing sequence of approximations of the domain. 2. Under strong spatial mixing (exponential decay of correlations even near boundary pinnings) in expectation, we prove a full Poincar\'e inequality, implying exponential relaxation to equilibrium and $N^{o(1)}$-mixing time. Note by way of example, both weak and strong spatial mixing hold at any temperature, provided the external fields are strong enough. Our proofs combine a stochastic localization technique which has the effect of increasing the variance of the field, with a field-dependent coarse graining which controls the resulting sub-critical percolation process of sites with weak fields.
