Showing papers in "Annals of Probability in 2020"

From nonlinear Fokker–Planck equations to solutions of distribution dependent SDE

Abstract: We construct weak solutions to the McKean–Vlasov SDE \begin{equation*}dX(t)=b\biggl(X(t),\frac{d{\mathcal{L}}_{X(t)}}{dx}\bigl(X(t)\bigr)\biggr)\,dt+\sigma\biggl(X(t),\frac{d{\mathcal{L}}_{X(t)}}{dt}\bigl(X(t)\bigr)\biggr)\,dW(t)\end{equation*} on ${\mathbb{R}}^{d}$ for possibly degenerate diffusion matrices $\sigma$ with $X(0)$ having a given law, which has a density with respect to Lebesgue measure, $dx$. Here, ${\mathcal{L}}_{X(t)}$ denotes the law of $X(t)$. Our approach is to first solve the corresponding nonlinear Fokker–Planck equations and then use the well-known superposition principle to obtain weak solutions of the above SDE.

Algorithmic thresholds for tensor PCA

Abstract: We study the algorithmic thresholds for principal component analysis of Gaussian $k$-tensors with a planted rank-one spike, via Langevin dynamics and gradient descent. In order to efficiently recover the spike from natural initializations, the signal-to-noise ratio must diverge in the dimension. Our proof shows that the mechanism for the success/failure of recovery is the strength of the “curvature” of the spike on the maximum entropy region of the initial data. To demonstrate this, we study the dynamics on a generalized family of high-dimensional landscapes with planted signals, containing the spiked tensor models as specific instances. We identify thresholds of signal-to-noise ratios above which order 1 time recovery succeeds; in the case of the spiked tensor model, these match the thresholds conjectured for algorithms such as approximate message passing. Below these thresholds, where the curvature of the signal on the maximal entropy region is weak, we show that recovery from certain natural initializations takes at least stretched exponential time. Our approach combines global regularity estimates for spin glasses with pointwise estimates to study the recovery problem by a perturbative approach.

From the master equation to mean field game limit theory: Large deviations and concentration of measure

Abstract: We study a sequence of symmetric $n$-player stochastic differential games driven by both idiosyncratic and common sources of noise, in which players interact with each other through their empirical distribution. The unique Nash equilibrium empirical measure of the $n$-player game is known to converge, as $n$ goes to infinity, to the unique equilibrium of an associated mean field game. Under suitable regularity conditions, in the absence of common noise, we complement this law of large numbers result with nonasymptotic concentration bounds for the Wasserstein distance between the $n$-player Nash equilibrium empirical measure and the mean field equilibrium. We also show that the sequence of Nash equilibrium empirical measures satisfies a weak large deviation principle, which can be strengthened to a full large deviation principle only in the absence of common noise. For both sets of results, we first use the master equation, an infinite-dimensional partial differential equation that characterizes the value function of the mean field game, to construct an associated McKean–Vlasov interacting $n$-particle system that is exponentially close to the Nash equilibrium dynamics of the $n$-player game for large $n$, by refining estimates obtained in our companion paper. Then we establish a weak large deviation principle for McKean–Vlasov systems in the presence of common noise. In the absence of common noise, we upgrade this to a full large deviation principle and obtain new concentration estimates for McKean–Vlasov systems. Finally, in two specific examples that do not satisfy the assumptions of our main theorems, we show how to adapt our methodology to establish large deviations and concentration results.

On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients

Abstract: We develop a perturbation theory for stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs). In particular, we estimate the $L^{p}$-distance between the solution process of an SDE and an arbitrary Ito process, which we view as a perturbation of the solution process of the SDE, by the $L^{q}$-distances of the differences of the local characteristics for suitable $p,q>0$. As one application of the developed perturbation theory, we establish strong convergence rates for numerical approximations of a class of SODEs with nonglobally monotone coefficients. As another application of the developed perturbation theory, we prove strong convergence rates for spatial spectral Galerkin approximations of solutions of semilinear SPDEs with nonglobally monotone nonlinearities including Cahn–Hilliard–Cook-type equations and stochastic Burgers equations. Further applications of the developed perturbation theory include regularity analyses of solutions of SDEs with respect to their initial values as well as small-noise analyses for ordinary and partial differential equations.

Averaging dynamics driven by fractional Brownian motion

Abstract: We consider slow/fast systems where the slow system is driven by fractional Brownian motion with Hurst parameter $H>{\frac{1}{2}}$. We show that unlike in the case $H={\frac{1}{2}}$, convergence to the averaged solution takes place in probability and the limiting process solves the ‘naively’ averaged equation. Our proof strongly relies on the recently obtained stochastic sewing lemma.

The Riemann zeta function and Gaussian multiplicative chaos: Statistics on the critical line

Abstract: We prove that if $\omega $ is uniformly distributed on $[0,1]$, then as $T\to \infty $, $t\mapsto \zeta (i\omega T+it+1/2)$ converges to a nontrivial random generalized function, which in turn is identified as a product of a very well-behaved random smooth function and a random generalized function known as a complex Gaussian multiplicative chaos distribution. This demonstrates a novel rigorous connection between probabilistic number theory and the theory of multiplicative chaos—the latter is known to be connected to various branches of modern probability theory and mathematical physics. We also investigate the statistical behavior of the zeta function on the mesoscopic scale. We prove that if we let $\delta _{T}$ approach zero slowly enough as $T\to \infty $, then $t\mapsto \zeta (1/2+i\delta _{T}t+i\omega T)$ is asymptotically a product of a divergent scalar quantity suggested by Selberg’s central limit theorem and a strictly Gaussian multiplicative chaos. We also prove a similar result for the characteristic polynomial of a Haar distributed random unitary matrix, where the scalar quantity is slightly different but the multiplicative chaos part is identical. This says that up to scalar multiples, the zeta function and the characteristic polynomial of a Haar distributed random unitary matrix have an identical distribution on the mesoscopic scale.

An almost sure KPZ relation for SLE and Brownian motion

Abstract: The peanosphere construction of Duplantier, Miller and Sheffield provides a means of representing a $\gamma $-Liouville quantum gravity (LQG) surface, $\gamma \in (0,2)$, decorated with a space-filling form of Schramm’s $\mathrm{SLE}_{\kappa }$, $\kappa =16/\gamma^{2}\in (4,\infty)$, $\eta $ as a gluing of a pair of trees which are encoded by a correlated two-dimensional Brownian motion $Z$. We prove a KPZ-type formula which relates the Hausdorff dimension of any Borel subset $A$ of the range of $\eta $, which can be defined as a function of $\eta $ (modulo time parameterization) to the Hausdorff dimension of the corresponding time set $\eta^{-1}(A)$. This result serves to reduce the problem of computing the Hausdorff dimension of any set associated with an $\mathrm{SLE}$, $\mathrm{CLE}$ or related processes in the interior of a domain to the problem of computing the Hausdorff dimension of a certain set associated with a Brownian motion. For many natural examples, the associated Brownian motion set is well known. As corollaries, we obtain new proofs of the Hausdorff dimensions of the $\mathrm{SLE}_{\kappa}$ curve for $\kappa eq4$; the double points and cut points of $\mathrm{SLE}_{\kappa }$ for $\kappa >4$; and the intersection of two flow lines of a Gaussian free field. We obtain the Hausdorff dimension of the set of $m$-tuple points of space-filling $\mathrm{SLE}_{\kappa }$ for $\kappa >4$ and $m\geq 3$ by computing the Hausdorff dimension of the so-called $(m-2)$-tuple $\pi /2$-cone times of a correlated planar Brownian motion.

The endpoint distribution of directed polymers

Abstract: Probabilistic models of directed polymers in random environment have received considerable attention in recent years. Much of this attention has focused on integrable models. In this paper, we introduce some new computational tools that do not require integrability. We begin by defining a new kind of abstract limit object, called “partitioned subprobability measure,” to describe the limits of endpoint distributions of directed polymers. Inspired by a recent work of Mukherjee and Varadhan on large deviations of the occupation measure of Brownian motion, we define a suitable topology on the space of partitioned subprobability measures and prove that this topology is compact. Then using a variant of the cavity method from the theory of spin glasses, we show that any limit law of a sequence of endpoint distributions must satisfy a fixed point equation on this abstract space, and that the limiting free energy of the model can be expressed as the solution of a variational problem over the set of fixed points. As a first application of the theory, we prove that in an environment with finite exponential moment, the endpoint distribution is asymptotically purely atomic if and only if the system is in the low temperature phase. The analogous result for a heavy-tailed environment was proved by Vargas in 2007. As a second application, we prove a subsequential version of the longstanding conjecture that in the low temperature phase, the endpoint distribution is asymptotically localized in a region of stochastically bounded diameter. All our results hold in arbitrary dimensions, and make no use of integrability.

The two-dimensional KPZ equation in the entire subcritical regime

Abstract: We consider the KPZ equation in space dimension $2$ driven by space-time white noise We showed in previous work that if the noise is mollified in space on scale $\varepsilon $ and its strength is scaled as $\hat{\beta }/\sqrt{|\log \varepsilon |}$, then a transition occurs with explicit critical point $\hat{\beta }_{c}=\sqrt{2\pi }$ Recently Chatterjee and Dunlap showed that the solution admits subsequential scaling limits as $\varepsilon \downarrow 0$, for sufficiently small $\hat{\beta }$ We prove here that the limit exists in the entire subcritical regime $\hat{\beta }\in (0,\hat{\beta }_{c})$ and we identify it as the solution of an additive stochastic heat equation, establishing so-called Edwards–Wilkinson fluctuations The same result holds for the directed polymer model in random environment in space dimension $2$

Convergence of transport noise to Ornstein–Uhlenbeck for 2D Euler equations under the enstrophy measure

Abstract: We consider the vorticity form of the 2D Euler equations which is perturbed by a suitable transport type noise and has white noise initial condition. It is shown that stationary solutions of this equation converge to the unique stationary solution of the 2D Navier–Stokes equation driven by the space-time white noise.

Transition from Tracy–Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erdős–Rényi graphs

Abstract: We consider the statistics of the extreme eigenvalues of sparse random matrices, a class of random matrices that includes the normalized adjacency matrices of the Erdős–Renyi graph $G(N,p)$. Tracy–Widom fluctuations of the extreme eigenvalues for $p\gg N^{-2/3}$ was proved in (Probab. Theory Related Fields 171 (2018) 543–616; Comm. Math. Phys. 314 (2012) 587–640). We prove that there is a crossover in the behavior of the extreme eigenvalues at $p\sim N^{-2/3}$. In the case that $N^{-7/9}\ll p\ll N^{-2/3}$, we prove that the extreme eigenvalues have asymptotically Gaussian fluctuations. Under a mean zero condition and when $p=CN^{-2/3}$, we find that the fluctuations of the extreme eigenvalues are given by a combination of the Gaussian and the Tracy–Widom distribution. These results show that the eigenvalues at the edge of the spectrum of sparse Erdős–Renyi graphs are less rigid than those of random $d$-regular graphs (Bauerschmidt et al. (2019)) of the same average degree.

Continuous Breuer–Major theorem: Tightness and nonstationarity

Abstract: Let $Y=(Y(t))_{t\geq 0}$ be a zero-mean Gaussian stationary process with covariance function $\rho :\mathbb{R}\to \mathbb{R}$ satisfying $\rho (0)=1$. Let $f:\mathbb{R}\to \mathbb{R}$ be a square-integrable function with respect to the standard Gaussian measure, and suppose the Hermite rank of $f$ is $d\geq 1$. If $\int_{\mathbb{R}}|\rho (s)|^{d}\,ds<\infty $, then the celebrated Breuer–Major theorem (in its continuous version) asserts that the finite-dimensional distributions of $Z_{\varepsilon }:=\sqrt{\varepsilon }\int_{0}^{\cdot /\varepsilon }f(Y(s))\,ds$ converge to those of $\sigma W$ as $\varepsilon \to 0$, where $W$ is a standard Brownian motion and $\sigma $ is some explicit constant. Since its first appearance in 1983, this theorem has become a crucial probabilistic tool in different areas, for instance in signal processing or in statistical inference for fractional Gaussian processes. The goal of this paper is twofold. First, we investigate the tightness in the Breuer–Major theorem. Surprisingly, this problem did not receive a lot of attention until now, and the best available condition due to Ben Hariz [J. Multivariate Anal. 80 (2002) 191–216] is neither arguably very natural, nor easy-to-check in practice. In contrast, our condition very simple, as it only requires that $|f|^{p}$ must be integrable with respect to the standard Gaussian measure for some $p$ strictly bigger than 2. It is obtained by means of the Malliavin calculus, in particular Meyer inequalities. Second, and motivated by a problem of geometrical nature, we extend the continuous Breuer–Major theorem to the notoriously difficult case of self-similar Gaussian processes which are not necessarily stationary. An application to the fluctuations associated with the length process of a regularized version of the bifractional Brownian motion concludes the paper.

Dimers and imaginary geometry

Abstract: We show that the winding of the branches in a uniform spanning tree on a planar graph converge in the limit of fine mesh size to a Gaussian free field. The result holds assuming only convergence of simple random walk to Brownian motion and a Russo-Seymour-Welsh type crossing estimate, thereby establishing a strong form of universality. As an application, we prove universality of the fluctuations of the height function associated to the dimer model, in several situations. The proof relies on a connection to imaginary geometry, where the scaling limit of a uniform spanning tree is viewed as a set of flow lines associated to a Gaussian free field. In particular, we obtain an explicit construction of the a.s. unique Gaussian free field coupled to a continuum uniform spanning tree in this way, which is of independent interest.

Large deviations for the largest eigenvalue of Rademacher matrices

Abstract: In this article, we consider random Wigner matrices, that is, symmetric matrices such that the subdiagonal entries of $X_{n}$ are independent, centered and with variance one except on the diagonal where the entries have variance two. We prove that, under some suitable hypotheses on the laws of the entries, the law of the largest eigenvalue satisfies a large deviation principle with the same rate function as in the Gaussian case. The crucial assumption is that the Laplace transform of the entries must be bounded above by the Laplace transform of a centered Gaussian variable with same variance. This is satisfied by the Rademacher law and the uniform law on $[-\sqrt{3},\sqrt{3}]$. We extend our result to complex entries Wigner matrices and Wishart matrices.

Locality of the critical probability for transitive graphs of exponential growth

Abstract: Around 2008, Schramm conjectured that the critical probabilities for Bernoulli bond percolation satisfy the following continuity property: If $(G_{n})_{n\geq 1}$ is a sequence of transitive graphs converging locally to a transitive graph $G$ and $\limsup_{n\to \infty }p_{c}(G_{n}) 1$ and $M<\infty $, there exist positive constants $C=C(g,M)$ and $\delta =\delta (g,M)$ such that if $G$ is a transitive unimodular graph with degree at most $M$ and growth $\operatorname{gr}(G):=\inf_{r\geq 1}|B(o,r)|^{1/r}\geq g$, then \[\mathbf{P}_{p_{c}}\bigl(\vert K_{o}\vert \geq n\bigr)\leq Cn^{-\delta }\] for every $n\geq 1$, where $K_{o}$ is the cluster of the root vertex $o$ The proof of this inequality makes use of new universal bounds on the probabilities of certain two-arm events, which hold for every unimodular transitive graph

Constructing a solution of the $(2+1)$-dimensional KPZ equation

Abstract: The $(d+1)$-dimensional KPZ equation is the canonical model for the growth of rough $d$-dimensional random surfaces. A deep mathematical understanding of the KPZ equation for $d=1$ has been achieved in recent years, and the case $d\ge 3$ has also seen some progress. The most physically relevant case of $d=2$, however, is not very well understood mathematically, largely due to the renormalization that is required: in the language of renormalization group analysis, the $d=2$ case is neither ultraviolet superrenormalizable like the $d=1$ case nor infrared superrenormalizable like the $d\ge 3$ case. Moreover, unlike in $d=1$, the Cole–Hopf transform is not directly usable in $d=2$ because solutions to the multiplicative stochastic heat equation are distributions rather than functions. In this article, we show the existence of subsequential scaling limits as $\varepsilon \to 0$ of Cole–Hopf solutions of the $(2+1)$-dimensional KPZ equation with white noise mollified to spatial scale $\varepsilon $ and nonlinearity multiplied by the vanishing factor $|\log \varepsilon |^{-\frac{1}{2}}$. We also show that the scaling limits obtained in this way do not coincide with solutions to the linearized equation, meaning that the nonlinearity has a nonvanishing effect. We thus propose our scaling limit as a notion of KPZ evolution in $2+1$ dimensions.

The distribution of Gaussian multiplicative chaos on the unit interval

Abstract: We consider a subcritical Gaussian multiplicative chaos (GMC) measure defined on the unit interval $[0,1]$ and prove an exact formula for the fractional moments of the total mass of this measure. Our formula includes the case where log-singularities (also called insertion points) are added in $0$ and $1$, the most general case predicted by the Selberg integral. The idea to perform this computation is to introduce certain auxiliary functions resembling holomorphic observables of conformal field theory that will be solutions of hypergeometric equations. Solving these equations then provides nontrivial relations that completely determine the moments we wish to compute. We also include a detailed discussion of the so-called reflection coefficients appearing in tail expansions of GMC measures and in Liouville theory. Our theorem provides an exact value for one of these coefficients. Lastly, we mention some additional applications to small deviations for GMC measures, to the behavior of the maximum of the log-correlated field on the interval and to random hermitian matrices.

Mean field systems on networks, with singular interaction through hitting times

Abstract: Building on the line of work (Ann. Appl. Probab. 25 (2015) 2096–2133; Stochastic Process. Appl. 125 (2015) 2451–2492; Ann. Appl. Probab. 29 (2019) 89–129; Arch. Ration. Mech. Anal. 233 (2019) 643–699; Ann. Appl. Probab. 29 (2019) 2338–2373; Finance Stoch. 23 (2019) 535–594), we continue the study of particle systems with singular interaction through hitting times. In contrast to the previous research, we (i) consider very general driving processes and interaction functions, (ii) allow for inhomogeneous connection structures and (iii) analyze a game in which the particles determine their connections strategically. Hereby, we uncover two completely new phenomena. First, we characterize the “times of fragility” of such systems (e.g., the times when a macroscopic part of the population defaults or gets infected simultaneously, or when the neuron cells “synchronize”) explicitly in terms of the dynamics of the driving processes, the current distribution of the particles’ values and the topology of the underlying network (represented by its Perron–Frobenius eigenvalue). Second, we use such systems to describe a dynamic credit-network game and show that, in equilibrium, the system regularizes, that is, the times of fragility never occur, as the particles avoid them by adjusting their connections strategically. Two auxiliary mathematical results, useful in their own right, are uncovered during our investigation: a generalization of Schauder’s fixed-point theorem for the Skorokhod space with the M1 topology, and the application of max-plus algebra to the equilibrium version of the network flow problem.

Nonlinear large deviation bounds with applications to Wigner matrices and sparse Erdős–Rényi graphs

Abstract: We prove general nonlinear large deviation estimates similar to Chatterjee–Dembo’s original bounds, except that we do not require any second order smoothness. Our approach relies on convex analysis arguments and is valid for a broad class of distributions. Our results are then applied in three different setups. Our first application consists in the mean-field approximation of the partition function of the Ising model under an optimal assumption on the spectra of the adjacency matrices of the sequence of graphs. Next, we apply our general large deviation bound to investigate the large deviation of the traces of powers of Wigner matrices with sub-Gaussian entries and the upper tail of cycles counts in sparse Erdős–Renyi graphs down to the sparsity threshold $n^{-1/2}$.

Correlated random matrices: Band rigidity and edge universality

Abstract: We prove edge universality for a general class of correlated real symmetric or complex Hermitian Wigner matrices with arbitrary expectation. Our theorem also applies to internal edges of the self-consistent density of states. In particular, we establish a strong form of band rigidity which excludes mismatches between location and label of eigenvalues close to internal edges in these general models.

Quenched invariance principle for random walks among random degenerate conductances

Abstract: We consider the random conductance model in a stationary and ergodic environment. Under suitable moment conditions on the conductances and their inverse, we prove a quenched invariance principle for the random walk among the random conductances. The moment conditions improve earlier results of Andres, Deuschel and Slowik (Ann. Probab. 43 (2015) 1866–1891) and are the minimal requirement to ensure that the corrector is sublinear everywhere. The key ingredient is an essentially optimal deterministic local boundedness result for finite difference equations in divergence form.

Entrance and exit at infinity for stable jump diffusions

Abstract: In his seminal work from the 1950s, William Feller classified all one-dimensional diffusions on −∞≤a

A covariance formula for topological events of smooth Gaussian fields

Abstract: We derive a covariance formula for the class of ‘topological events’ of smooth Gaussian fields on manifolds; these are events that depend only on the topology of the level sets of the field, for example, (i) crossing events for level or excursion sets, (ii) events measurable with respect to the number of connected components of level or excursion sets of a given diffeomorphism class and (iii) persistence events. As an application of the covariance formula, we derive strong mixing bounds for topological events, as well as lower concentration inequalities for additive topological functionals (e.g., the number of connected components) of the level sets that satisfy a law of large numbers. The covariance formula also gives an alternate justification of the Harris criterion, which conjecturally describes the boundary of the percolation university class for level sets of stationary Gaussian fields. Our work is inspired by (Ann. Inst. Henri Poincare Probab. Stat. 55 (2019) 1679–1711), in which a correlation inequality was derived for certain topological events on the plane, as well as by (Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996) Amer. Math. Soc.), in which a similar covariance formula was established for finite-dimensional Gaussian vectors.

Martingale Benamou–Brenier: A probabilistic perspective

Abstract: In classical optimal transport, the contributions of Benamou–Brenier and McCann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas. We suggest a Benamou–Brenier type formulation of the martingale transport problem for given $d$-dimensional distributions $\mu $, $ u $ in convex order. The unique solution $M^{*}=(M_{t}^{*})_{t\in [0,1]}$ of this problem turns out to be a Markov-martingale which has several notable properties: In a specific sense it mimics the movement of a Brownian particle as closely as possible subject to the conditions $M^{*}_{0}\sim \mu $, $M^{*}_{1}\sim u $. Similar to McCann’s displacement-interpolation, $M^{*}$ provides a time-consistent interpolation between $\mu $ and $ u $. For particular choices of the initial and terminal law, $M^{*}$ recovers archetypical martingales such as Brownian motion, geometric Brownian motion, and the Bass martingale. Furthermore, it yields a natural approximation to the local vol model and a new approach to Kellerer’s theorem. This article is parallel to the work of Huesmann–Trevisan, who consider a related class of problems from a PDE-oriented perspective.

On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates

Abstract: We show that, for a strongly local, regular symmetric Dirichlet form over a complete, locally compact geodesic metric space, full off-diagonal heat kernel estimates with walk dimension strictly larger than two (sub-Gaussian estimates) imply the singularity of the energy measures with respect to the symmetric measure, verifying a conjecture by M. T. Barlow in (Contemp. Math. 338 (2003) 11–40). We also prove that in the contrary case of walk dimension two, that is, where full off-diagonal Gaussian estimates of the heat kernel hold, the symmetric measure and the energy measures are mutually absolutely continuous in the sense that a Borel subset of the state space has measure zero for the symmetric measure if and only if it has measure zero for the energy measures of all functions in the domain of the Dirichlet form.

Planar Brownian motion and Gaussian multiplicative chaos

Abstract: We construct the analogue of Gaussian multiplicative chaos measures for the local times of planar Brownian motion by exponentiating the square root of the local times of small circles. We also consider a flat measure supported on points whose local time is within a constant of the desired thickness level and show a simple relation between the two objects. Our results extend those of (Ann. Probab. 22 (1994) 566–625), and in particular, cover the entire $L^{1}$-phase or subcritical regime. These results allow us to obtain a nondegenerate limit for the appropriately rescaled size of thick points, thereby considerably refining estimates of (Acta Math. 186 (2001) 239–270).

Elliptic stochastic quantization

Abstract: We prove an explicit formula for the law in zero of the solution of a class of elliptic SPDE in $\mathbb{R}^{2}$. This formula is the simplest instance of dimensional reduction, discovered in the physics literature by Parisi and Sourlas (Phys. Rev. Lett. 43 (1979) 744–745), which links the law of an elliptic SPDE in $d+2$ dimension with a Gibbs measure in $d$ dimensions. This phenomenon is similar to the relation between a $\mathbb{R}^{d+1}$ dimensional parabolic SPDE and its $\mathbb{R}^{d}$ dimensional invariant measure. As such, dimensional reduction of elliptic SPDEs can be considered a sort of elliptic stochastic quantisation procedure in the sense of Nelson (Phys. Rev. 150 (1966) 1079–1085) and Parisi and Wu (Sci. Sin. 24 (1981) 483–496). Our proof uses in a fundamental way the representation of the law of the SPDE as a supersymmetric quantum field theory. Dimensional reduction for the supersymmetric theory was already established by Klein et al. (Comm. Math. Phys. 94 (1984) 459–482). We fix a subtle gap in their proof and also complete the dimensional reduction picture by providing the link between the elliptic SPDE and the supersymmetric model. Even in our $d=0$ context the arguments are nontrivial and a nonsupersymmetric, elementary proof seems only to be available in the Gaussian case.

The CLT in high dimensions: Quantitative bounds via martingale embedding

Abstract: We introduce a new method for obtaining quantitative convergence rates for the central limit theorem (CLT) in a high-dimensional setting. Using our method, we obtain several new bounds for convergence in transportation distance and entropy, and in particular: (a) We improve the best known bound, obtained by the third named author (Probab. Theory Related Fields 170 (2018) 821–845), for convergence in quadratic Wasserstein transportation distance for bounded random vectors; (b) we derive the first nonasymptotic convergence rate for the entropic CLT in arbitrary dimension, for general log-concave random vectors (this adds to (Ann. Inst. Henri Poincare Probab. Stat. 55 (2019) 777–790), where a finite Fisher information is assumed); (c) we give an improved bound for convergence in transportation distance under a log-concavity assumption and improvements for both metrics under the assumption of strong log-concavity. Our method is based on martingale embeddings and specifically on the Skorokhod embedding constructed in (Ann. Inst. Henri Poincare Probab. Stat. 52 (2016) 1259–1280).

The exclusion process mixes (almost) faster than independent particles

Abstract: Oliveira conjectured that the order of the mixing time of the exclusion process with k-particles on an arbitrary n-vertex graph is at most that of the mixing-time of k independent particles. We verify this up to a constant factor for d-regular graphs when each edge rings at rate 1/d in various cases: (1) when d = Ω(logn/k n), (2) when gap := the spectral-gap of a single walk is O(1/ log4 n) and k > n Ω(1) , (3) when k ≍ n a for some constant 0 < a < 1. In these cases our analysis yields a probabilistic proof of a weaker version of Aldous’ famous spectral-gap conjecture (resolved by Caputo et al.). We also prove a general bound of O(log n log log n/gap), which is within a log log n factor from Oliveira’s conjecture when k > n Ω(1). As applications we get new mixing bounds: (a) O(log n log log n) for expanders, (b) order d log(dk) for the hypercube {0, 1} d , (c) order (Diameter)2 log k for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus.

The maximum of the four-dimensional membrane model

Abstract: We show that the centred maximum of the four-dimensional membrane model on a box of sidelength $N$ converges in distribution. To do so, we use a criterion of Ding, Roy and Zeitouni (Ann. Probab. 45 (2017) 3886–3928) and prove sharp estimates for the Green’s function of the discrete Bilaplacian. These estimates are the main contribution of this work and might also be of independent interest. To derive them, we use estimates for the approximation quality of finite difference schemes as well as results for the Green’s function of the continuous Bilaplacian.