The Annals of Probability Articles (Project Euclid)
http://projecteuclid.org/euclid.aop
The latest articles from The Annals of Probability on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTWed, 16 Mar 2011 09:23 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
http://projecteuclid.org/
Uniform convergence of Vapnik–Chervonenkis classes under ergodic sampling
http://projecteuclid.org/euclid.aop/1278593952
<strong>Terrence M. Adams</strong>, <strong>Andrew B. Nobel</strong><p><strong>Source: </strong>Ann. Probab., Volume 38, Number 4, 1345--1367.</p><p><strong>Abstract:</strong><br/>
We show that if $\mathcal{X}$ is a complete separable metric space and $\mathcal{C}$ is a countable family of Borel subsets of $\mathcal{X}$ with finite VC dimension, then, for every stationary ergodic process with values in $\mathcal{X}$ , the relative frequencies of sets $C\in\mathcal{C}$ converge uniformly to their limiting probabilities. Beyond ergodicity, no assumptions are imposed on the sampling process, and no regularity conditions are imposed on the elements of $\mathcal{C}$ . The result extends existing work of Vapnik and Chervonenkis, among others, who have studied uniform convergence for i.i.d. and strongly mixing processes. Our method of proof is new and direct: it does not rely on symmetrization techniques, probability inequalities or mixing conditions. The uniform convergence of relative frequencies for VC-major and VC-graph classes of functions under ergodic sampling is established as a corollary of the basic result for sets.
</p>projecteuclid.org/euclid.aop/1278593952_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTScaling limit of two-component interacting Brownian motionshttps://projecteuclid.org/euclid.aop/1528876821<strong>Insuk Seo</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2038--2063.</p><p><strong>Abstract:</strong><br/>
This paper presents our study of the asymptotic behavior of a two-component system of Brownian motions undergoing certain form of singular interactions. In particular, the system is a combination of two different types of particles and the mechanical properties and the interaction parameters depend on the corresponding type of particles. We prove that the hydrodynamic limit of the empirical densities of two types is the solution of a partial differential equation known as the Maxwell–Stefan equation.
</p>projecteuclid.org/euclid.aop/1528876821_20180613040038Wed, 13 Jun 2018 04:00 EDTLarge excursions and conditioned laws for recursive sequences generated by random matriceshttps://projecteuclid.org/euclid.aop/1528876822<strong>Jeffrey F. Collamore</strong>, <strong>Sebastian Mentemeier</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2064--2120.</p><p><strong>Abstract:</strong><br/>
We study the large exceedance probabilities and large exceedance paths of the recursive sequence $V_{n}=M_{n}V_{n-1}+Q_{n}$, where $\{(M_{n},Q_{n})\}$ is an i.i.d. sequence, and $M_{1}$ is a $d\times d$ random matrix and $Q_{1}$ is a random vector, both with nonnegative entries. We impose conditions which guarantee the existence of a unique stationary distribution for $\{V_{n}\}$ and a Cramér-type condition for $\{M_{n}\}$. Under these assumptions, we characterize the distribution of the first passage time $T_{u}^{A}:=\inf\{n:V_{n}\in uA\}$, where $A$ is a general subset of $\mathbb{R}^{d}$, exhibiting that $T_{u}^{A}/u^{\alpha}$ converges to an exponential law for a certain $\alpha>0$. In the process, we revisit and refine classical estimates for $\mathbb{P}(V\in uA)$, where $V$ possesses the stationary law of $\{V_{n}\}$. Namely, for $A\subset\mathbb{R}^{d}$, we show that $\mathbb{P}(V\in uA)\sim C_{A}u^{-\alpha}$ as $u\to\infty$, providing, most importantly, a new characterization of the constant $C_{A}$. As a simple consequence of these estimates, we also obtain an expression for the extremal index of $\{|V_{n}|\}$. Finally, we describe the large exceedance paths via two conditioned limit theorems showing, roughly, that $\{V_{n}\}$ follows an exponentially-shifted Markov random walk, which we identify. We thereby generalize results from the theory of classical random walk to multivariate recursive sequences.
</p>projecteuclid.org/euclid.aop/1528876822_20180613040038Wed, 13 Jun 2018 04:00 EDTPhase transition for the Once-reinforced random walk on $\mathbb{Z}^{d}$-like treeshttps://projecteuclid.org/euclid.aop/1528876823<strong>Daniel Kious</strong>, <strong>Vladas Sidoravicius</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2121--2133.</p><p><strong>Abstract:</strong><br/>
In this short paper, we consider the Once-reinforced random walk with reinforcement parameter $a$ on trees with bounded degree which are transient for the simple random walk. On each of these trees, we prove that there exists an explicit critical parameter $a_{0}$ such that the Once-reinforced random walk is almost surely recurrent if $a>a_{0}$ and almost surely transient if $a<a_{0}$. This provides the first examples of phase transition for the Once-reinforced random walk.
</p>projecteuclid.org/euclid.aop/1528876823_20180613040038Wed, 13 Jun 2018 04:00 EDTThe Brownian limit of separable permutationshttps://projecteuclid.org/euclid.aop/1528876824<strong>Frédérique Bassino</strong>, <strong>Mathilde Bouvel</strong>, <strong>Valentin Féray</strong>, <strong>Lucas Gerin</strong>, <strong>Adeline Pierrot</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2134--2189.</p><p><strong>Abstract:</strong><br/>
We study uniform random permutations in an important class of pattern-avoiding permutations: the separable permutations. We describe the asymptotics of the number of occurrences of any fixed given pattern in such a random permutation in terms of the Brownian excursion. In the recent terminology of permutons, our work can be interpreted as the convergence of uniform random separable permutations towards a “Brownian separable permuton”.
</p>projecteuclid.org/euclid.aop/1528876824_20180613040038Wed, 13 Jun 2018 04:00 EDTCritical density of activated random walks on transitive graphshttps://projecteuclid.org/euclid.aop/1528876825<strong>Alexandre Stauffer</strong>, <strong>Lorenzo Taggi</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2190--2220.</p><p><strong>Abstract:</strong><br/>
We consider the activated random walk model on general vertex-transitive graphs. A central question in this model is whether the critical density $\mu_{c}$ for sustained activity is strictly between 0 and 1. It was known that $\mu_{c}>0$ on $\mathbb{Z}^{d}$, $d\geq1$, and that $\mu_{c}<1$ on $\mathbb{Z}$ for small enough sleeping rate. We show that $\mu_{c}\to0$ as $\lambda\to0$ in all vertex-transitive transient graphs, implying that $\mu_{c}<1$ for small enough sleeping rate. We also show that $\mu_{c}<1$ for any sleeping rate in any vertex-transitive graph in which simple random walk has positive speed. Furthermore, we prove that $\mu_{c}>0$ in any vertex-transitive amenable graph, and that $\mu_{c}\in(0,1)$ for any sleeping rate on regular trees.
</p>projecteuclid.org/euclid.aop/1528876825_20180613040038Wed, 13 Jun 2018 04:00 EDTIndistinguishability of the components of random spanning forestshttps://projecteuclid.org/euclid.aop/1528876826<strong>Ádám Timár</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2221--2242.</p><p><strong>Abstract:</strong><br/>
We prove that the infinite components of the Free Uniform Spanning Forest (FUSF) of a Cayley graph are indistinguishable by any invariant property, given that the forest is different from its wired counterpart. Similar result is obtained for the Free Minimal Spanning Forest (FMSF). We also show that with the above assumptions there can only be 0, 1 or infinitely many components, which solves the problem for the FUSF of Caylay graphs completely. These answer questions by Benjamini, Lyons, Peres and Schramm for Cayley graphs, which have been open up to now. Our methods apply to a more general class of percolations, those satisfying “weak insertion tolerance”, and work beyond Cayley graphs, in the more general setting of unimodular random graphs.
</p>projecteuclid.org/euclid.aop/1528876826_20180613040038Wed, 13 Jun 2018 04:00 EDTWeak symmetric integrals with respect to the fractional Brownian motionhttps://projecteuclid.org/euclid.aop/1528876827<strong>Giulia Binotto</strong>, <strong>Ivan Nourdin</strong>, <strong>David Nualart</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2243--2267.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to establish the weak convergence, in the topology of the Skorohod space, of the $\nu$-symmetric Riemann sums for functionals of the fractional Brownian motion when the Hurst parameter takes the critical value $H=(4\ell+2)^{-1}$, where $\ell=\ell(\nu)\geq1$ is the largest natural number satisfying $\int_{0}^{1}\alpha^{2j}\nu(d\alpha)=\frac{1}{2j+1}$ for all $j=0,\ldots,\ell-1$. As a consequence, we derive a change-of-variable formula in distribution, where the correction term is a stochastic integral with respect to a Brownian motion that is independent of the fractional Brownian motion.
</p>projecteuclid.org/euclid.aop/1528876827_20180613040038Wed, 13 Jun 2018 04:00 EDTOn the spectral radius of a random matrix: An upper bound without fourth momenthttps://projecteuclid.org/euclid.aop/1528876828<strong>Charles Bordenave</strong>, <strong>Pietro Caputo</strong>, <strong>Djalil Chafaï</strong>, <strong>Konstantin Tikhomirov</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2268--2286.</p><p><strong>Abstract:</strong><br/>
Consider a square matrix with independent and identically distributed entries of zero mean and unit variance. It is well known that if the entries have a finite fourth moment, then, in high dimension, with high probability, the spectral radius is close to the square root of the dimension. We conjecture that this holds true under the sole assumption of zero mean and unit variance. In other words, that there are no outliers in the circular law. In this work, we establish the conjecture in the case of symmetrically distributed entries with a finite moment of order larger than two. The proof uses the method of moments combined with a novel truncation technique for cycle weights that might be of independent interest.
</p>projecteuclid.org/euclid.aop/1528876828_20180613040038Wed, 13 Jun 2018 04:00 EDTStochastic Airy semigroup through tridiagonal matriceshttps://projecteuclid.org/euclid.aop/1528876829<strong>Vadim Gorin</strong>, <strong>Mykhaylo Shkolnikov</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2287--2344.</p><p><strong>Abstract:</strong><br/>
We determine the operator limit for large powers of random symmetric tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for the Laplace transform of the Airy$_{\beta}$ process, which describes the largest eigenvalues in the $\beta$ ensembles of random matrix theory. Another consequence is a Feynman–Kac formula for the stochastic Airy operator of Edelman–Sutton and Ramirez–Rider–Virag.
As a side result, we find that the difference between the area underneath a standard Brownian excursion and one half of the integral of its squared local times is a Gaussian random variable.
</p>projecteuclid.org/euclid.aop/1528876829_20180613040038Wed, 13 Jun 2018 04:00 EDTOn the mixing time of Kac’s walk and other high-dimensional Gibbs samplers with constraintshttps://projecteuclid.org/euclid.aop/1528876830<strong>Natesh S. Pillai</strong>, <strong>Aaron Smith</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2345--2399.</p><p><strong>Abstract:</strong><br/>
Determining the total variation mixing time of Kac’s random walk on the special orthogonal group $\mathrm{SO}(n)$ has been a long-standing open problem. In this paper, we construct a novel non-Markovian coupling for bounding this mixing time. The analysis of our coupling entails controlling the smallest singular value of a certain random matrix with highly dependent entries. The dependence of the entries in our matrix makes it not amenable to existing techniques in random matrix theory. To circumvent this difficulty, we extend some recent bounds on the smallest singular values of matrices with independent entries to our setting. These bounds imply that the mixing time of Kac’s walk on the group $\mathrm{SO}(n)$ is between $C_{1}n^{2}$ and $C_{2}n^{4}\log(n)$ for some explicit constants $0<C_{1},C_{2}<\infty$, substantially improving on the bound of $O(n^{5}\log(n)^{2})$ in the preprint of Jiang [Jiang (2012)]. Our methods may also be applied to other high dimensional Gibbs samplers with constraints, and thus are of independent interest. In addition to giving analytical bounds on the mixing time, our approach allows us to compute rigorous estimates of the mixing time by simulating the eigenvalues of a random matrix.
</p>projecteuclid.org/euclid.aop/1528876830_20180613040038Wed, 13 Jun 2018 04:00 EDTErrata to “Distance covariance in metric spaces”https://projecteuclid.org/euclid.aop/1528876831<strong>Russell Lyons</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2400--2405.</p><p><strong>Abstract:</strong><br/>
We correct several statements and proofs in our paper, Ann. Probab. 41 , no. 5 (2013), 3284–3305.
</p>projecteuclid.org/euclid.aop/1528876831_20180613040038Wed, 13 Jun 2018 04:00 EDTRoots of random polynomials with coefficients of polynomial growthhttps://projecteuclid.org/euclid.aop/1535097632<strong>Yen Do</strong>, <strong>Oanh Nguyen</strong>, <strong>Van Vu</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2407--2494.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coefficients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these polynomials, even when the coefficients are not explicit. Our results also hold for series; in particular, we prove local universality for random hyperbolic series.
</p>projecteuclid.org/euclid.aop/1535097632_20180824040102Fri, 24 Aug 2018 04:01 EDTWell-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDEhttps://projecteuclid.org/euclid.aop/1535097633<strong>Benjamin Gess</strong>, <strong>Martina Hofmanová</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2495--2544.</p><p><strong>Abstract:</strong><br/>
We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full $L^{1}$ setting requiring no growth assumptions on the nonlinearities. In addition, we prove a comparison result and an $L^{1}$-contraction property for the solutions, generalizing the results obtained in [ Ann. Probab. 44 (2016) 1916–1955].
</p>projecteuclid.org/euclid.aop/1535097633_20180824040102Fri, 24 Aug 2018 04:01 EDTThree favorite sites occurs infinitely often for one-dimensional simple random walkhttps://projecteuclid.org/euclid.aop/1535097634<strong>Jian Ding</strong>, <strong>Jianfei Shen</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2545--2561.</p><p><strong>Abstract:</strong><br/>
For a one-dimensional simple random walk $(S_{t})$, for each time $t$ we say a site $x$ is a favorite site if it has the maximal local time. In this paper, we show that with probability 1 three favorite sites occurs infinitely often. Our work is inspired by Tóth [ Ann. Probab. 29 (2001) 484–503], and disproves a conjecture of Erdős and Révész [In Mathematical Structure—Computational Mathematics—Mathematical Modelling 2 (1984) 152–157] and of Tóth [ Ann. Probab. 29 (2001) 484–503].
</p>projecteuclid.org/euclid.aop/1535097634_20180824040102Fri, 24 Aug 2018 04:01 EDTZigzag diagrams and Martin boundaryhttps://projecteuclid.org/euclid.aop/1535097635<strong>Pierre Tarrago</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2562--2620.</p><p><strong>Abstract:</strong><br/>
We investigate the asymptotic behavior of random paths on a graded graph which describes the subword order for words in two letters. This graph, denoted by $\mathcal{Z}$, has been introduced by Viennot, who also discovered a remarkable bijection between paths on $\mathcal{Z}$ and sequences of permutations. Later on, Gnedin and Olshanski used this bijection to describe the set of Gibbs measures on this graph. Both authors also conjectured that the Martin boundary of $\mathcal{Z}$ should coincide with its minimal boundary. We give here a proof of this conjecture by describing the distribution of a large random path conditioned on having a prescribed endpoint. We also relate paths on the graph $\mathcal{Z}$ with paths on the Young lattice, and we finally give a central limit theorem for the Plancherel measure on the set of paths in $\mathcal{Z}$.
</p>projecteuclid.org/euclid.aop/1535097635_20180824040102Fri, 24 Aug 2018 04:01 EDTParacontrolled distributions and the 3-dimensional stochastic quantization equationhttps://projecteuclid.org/euclid.aop/1535097636<strong>Rémi Catellier</strong>, <strong>Khalil Chouk</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2621--2679.</p><p><strong>Abstract:</strong><br/>
We prove the existence and uniqueness of a local in time solution to the periodic $\Phi^{4}_{3}$ model of stochastic quantisation using the method of paracontrolled distributions introduced recently by M. Gubinelli, P. Imkeller and N. Perkowski in [ Forum Math., Pi 3 (2015) e6].
</p>projecteuclid.org/euclid.aop/1535097636_20180824040102Fri, 24 Aug 2018 04:01 EDTStable random fields indexed by finitely generated free groupshttps://projecteuclid.org/euclid.aop/1535097637<strong>Sourav Sarkar</strong>, <strong>Parthanil Roy</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2680--2714.</p><p><strong>Abstract:</strong><br/>
In this work, we investigate the extremal behaviour of left-stationary symmetric $\alpha$-stable (S$\alpha$S) random fields indexed by finitely generated free groups. We begin by studying the rate of growth of a sequence of partial maxima obtained by varying the indexing parameter of the field over balls of increasing size. This leads to a phase-transition that depends on the ergodic properties of the underlying nonsingular action of the free group but is different from what happens in the case of S$\alpha$S random fields indexed by $\mathbb{Z}^{d}$. The presence of this new dichotomy is confirmed by the study of a stable random field induced by the canonical action of the free group on its Furstenberg–Poisson boundary with the measure being Patterson–Sullivan. This field is generated by a conservative action but its maxima sequence grows as fast as the i.i.d. case contrary to what happens in the case of $\mathbb{Z}^{d}$. When the action of the free group is dissipative, we also establish that the scaled extremal point process sequence converges weakly to a novel class of point processes that we have termed as randomly thinned cluster Poisson processes . This limit too is very different from that in the case of a lattice.
</p>projecteuclid.org/euclid.aop/1535097637_20180824040102Fri, 24 Aug 2018 04:01 EDTRecursive construction of continuum random treeshttps://projecteuclid.org/euclid.aop/1535097638<strong>Franz Rembart</strong>, <strong>Matthias Winkel</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2715--2748.</p><p><strong>Abstract:</strong><br/>
We introduce a general recursive method to construct continuum random trees (CRTs) from independent copies of a random string of beads, that is, any random interval equipped with a random discrete probability measure, and from related structures. We prove the existence of these CRTs as a new application of the fixpoint method for recursive distribution equations formalised in high generality by Aldous and Bandyopadhyay.
We apply this recursive method to show the convergence to CRTs of various tree growth processes. We note alternative constructions of existing self-similar CRTs in the sense of Haas, Miermont and Stephenson, and we give for the first time constructions of random compact $\mathbb{R}$-trees that describe the genealogies of Bertoin’s self-similar growth fragmentations. In forthcoming work, we develop further applications to embedding problems for CRTs, providing a binary embedding of the stable line-breaking construction that solves an open problem of Goldschmidt and Haas.
</p>projecteuclid.org/euclid.aop/1535097638_20180824040102Fri, 24 Aug 2018 04:01 EDTControlled equilibrium selection in stochastically perturbed dynamicshttps://projecteuclid.org/euclid.aop/1535097639<strong>Ari Arapostathis</strong>, <strong>Anup Biswas</strong>, <strong>Vivek S. Borkar</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2749--2799.</p><p><strong>Abstract:</strong><br/>
We consider a dynamical system with finitely many equilibria and perturbed by small noise, in addition to being controlled by an “expensive” control. The controlled process is optimal for an ergodic criterion with a running cost that consists of the sum of the control effort and a penalty function on the state space. We study the optimal stationary distribution of the controlled process as the variance of the noise becomes vanishingly small. It is shown that depending on the relative magnitudes of the noise variance and the “running cost” for control, one can identify three regimes, in each of which the optimal control forces the invariant distribution of the process to concentrate near equilibria that can be characterized according to the regime. We also obtain moment bounds for the optimal stationary distribution. Moreover, we show that in the vicinity of the points of concentration the density of optimal stationary distribution approximates the density of a Gaussian, and we explicitly solve for its covariance matrix.
</p>projecteuclid.org/euclid.aop/1535097639_20180824040102Fri, 24 Aug 2018 04:01 EDTA new look at duality for the symbiotic branching modelhttps://projecteuclid.org/euclid.aop/1535097640<strong>Matthias Hammer</strong>, <strong>Marcel Ortgiese</strong>, <strong>Florian Völlering</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2800--2862.</p><p><strong>Abstract:</strong><br/>
The symbiotic branching model is a spatial population model describing the dynamics of two interacting types that can only branch if both types are present. A classical result for the underlying stochastic partial differential equation identifies moments of the solution via a duality to a system of Brownian motions with dynamically changing colors. In this paper, we revisit this duality and give it a new interpretation. This new approach allows us to extend the duality to the limit as the branching rate $\gamma$ is sent to infinity. This limit is particularly interesting since it captures the large scale behavior of the system. As an application of the duality, we can explicitly identify the $\gamma=\infty$ limit when the driving noises are perfectly negatively correlated. The limit is a system of annihilating Brownian motions with a drift that depends on the initial imbalance between the types.
</p>projecteuclid.org/euclid.aop/1535097640_20180824040102Fri, 24 Aug 2018 04:01 EDTAlternating arm exponents for the critical planar Ising modelhttps://projecteuclid.org/euclid.aop/1535097641<strong>Hao Wu</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2863--2907.</p><p><strong>Abstract:</strong><br/>
We derive the alternating arm exponents of the critical Ising model. We obtain six different patterns of alternating boundary arm exponents which correspond to the boundary conditions $(\ominus\oplus)$, $(\ominus\operatorname{free})$ and $(\operatorname{free}\operatorname{free})$, and the alternating interior arm exponents.
</p>projecteuclid.org/euclid.aop/1535097641_20180824040102Fri, 24 Aug 2018 04:01 EDTGaussian mixtures: Entropy and geometric inequalitieshttps://projecteuclid.org/euclid.aop/1535097642<strong>Alexandros Eskenazis</strong>, <strong>Piotr Nayar</strong>, <strong>Tomasz Tkocz</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2908--2945.</p><p><strong>Abstract:</strong><br/>
A symmetric random variable is called a Gaussian mixture if it has the same distribution as the product of two independent random variables, one being positive and the other a standard Gaussian random variable. Examples of Gaussian mixtures include random variables with densities proportional to $e^{-|t|^{p}}$ and symmetric $p$-stable random variables, where $p\in(0,2]$. We obtain various sharp moment and entropy comparison estimates for weighted sums of independent Gaussian mixtures and investigate extensions of the B-inequality and the Gaussian correlation inequality in the context of Gaussian mixtures. We also obtain a correlation inequality for symmetric geodesically convex sets in the unit sphere equipped with the normalized surface area measure. We then apply these results to derive sharp constants in Khinchine inequalities for vectors uniformly distributed on the unit balls with respect to $p$-norms and provide short proofs to new and old comparison estimates for geometric parameters of sections and projections of such balls.
</p>projecteuclid.org/euclid.aop/1535097642_20180824040102Fri, 24 Aug 2018 04:01 EDTThe survival probability of a critical multi-type branching process in i.i.d. random environmenthttps://projecteuclid.org/euclid.aop/1535097643<strong>E. Le Page</strong>, <strong>M. Peigné</strong>, <strong>C. Pham</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2946--2972.</p><p><strong>Abstract:</strong><br/>
We study the asymptotic behaviour of the probability of non-extinction of a critical multi-type Galton–Watson process in i.i.d. random environments by using limit theorems for products of positive random matrices. Under suitable assumptions, the survival probability is proportional to $1/\sqrt{n}$.
</p>projecteuclid.org/euclid.aop/1535097643_20180824040102Fri, 24 Aug 2018 04:01 EDTAiry point process at the liquid-gas boundaryhttps://projecteuclid.org/euclid.aop/1535097644<strong>Vincent Beffara</strong>, <strong>Sunil Chhita</strong>, <strong>Kurt Johansson</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2973--3013.</p><p><strong>Abstract:</strong><br/>
Domino tilings of the two-periodic Aztec diamond feature all of the three possible types of phases of random tiling models. These phases are determined by the decay of correlations between dominoes and are generally known as solid, liquid and gas. The liquid-solid boundary is easy to define microscopically and is known in many models to be described by the Airy process in the limit of a large random tiling. The liquid-gas boundary has no obvious microscopic description. Using the height function, we define a random measure in the two-periodic Aztec diamond designed to detect the long range correlations visible at the liquid-gas boundary. We prove that this random measure converges to the extended Airy point process. This indicates that, in a sense, the liquid-gas boundary should also be described by the Airy process.
</p>projecteuclid.org/euclid.aop/1535097644_20180824040102Fri, 24 Aug 2018 04:01 EDTPfaffian Schur processes and last passage percolation in a half-quadranthttps://projecteuclid.org/euclid.aop/1537862428<strong>Jinho Baik</strong>, <strong>Guillaume Barraquand</strong>, <strong>Ivan Corwin</strong>, <strong>Toufic Suidan</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3015--3089.</p><p><strong>Abstract:</strong><br/>
We study last passage percolation in a half-quadrant, which we analyze within the framework of Pfaffian Schur processes. For the model with exponential weights, we prove that the fluctuations of the last passage time to a point on the diagonal are either GSE Tracy–Widom distributed, GOE Tracy–Widom distributed or Gaussian, depending on the size of weights along the diagonal. Away from the diagonal, the fluctuations of passage times follow the GUE Tracy–Widom distribution. We also obtain a two-dimensional crossover between the GUE, GOE and GSE distribution by studying the multipoint distribution of last passage times close to the diagonal when the size of the diagonal weights is simultaneously scaled close to the critical point. We expect that this crossover arises universally in KPZ growth models in half-space. Along the way, we introduce a method to deal with diverging correlation kernels of point processes where points collide in the scaling limit.
</p>projecteuclid.org/euclid.aop/1537862428_20180925040105Tue, 25 Sep 2018 04:01 EDTPathwise uniqueness of the stochastic heat equation with spatially inhomogeneous white noisehttps://projecteuclid.org/euclid.aop/1537862429<strong>Eyal Neuman</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3090--3187.</p><p><strong>Abstract:</strong><br/>
We study the solutions of the stochastic heat equation driven by spatially inhomogeneous multiplicative white noise based on a fractal measure. We prove pathwise uniqueness for solutions of this equation when the noise coefficient is Hölder continuous of index $\gamma>1-\frac{\eta}{2(\eta+1)}$. Here $\eta\in(0,1)$ is a constant that defines the spatial regularity of the noise.
</p>projecteuclid.org/euclid.aop/1537862429_20180925040105Tue, 25 Sep 2018 04:01 EDTA quantitative central limit theorem for the Euler–Poincaré characteristic of random spherical eigenfunctionshttps://projecteuclid.org/euclid.aop/1537862432<strong>Valentina Cammarota</strong>, <strong>Domenico Marinucci</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3188--3228.</p><p><strong>Abstract:</strong><br/>
We establish here a quantitative central limit theorem (in Wasserstein distance) for the Euler–Poincaré characteristic of excursion sets of random spherical eigenfunctions in dimension 2. Our proof is based upon a decomposition of the Euler–Poincaré characteristic into different Wiener-chaos components: we prove that its asymptotic behaviour is dominated by a single term, corresponding to the chaotic component of order two. As a consequence, we show how the asymptotic dependence on the threshold level $u$ is fully degenerate, that is, the Euler–Poincaré characteristic converges to a single random variable times a deterministic function of the threshold. This deterministic function has a zero at the origin, where the variance is thus asymptotically of smaller order. We discuss also a possible unifying framework for the Lipschitz–Killing curvatures of the excursion sets for Gaussian spherical harmonics.
</p>projecteuclid.org/euclid.aop/1537862432_20180925040105Tue, 25 Sep 2018 04:01 EDTRepresentations and isomorphism identities for infinitely divisible processeshttps://projecteuclid.org/euclid.aop/1537862433<strong>Jan Rosiński</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3229--3274.</p><p><strong>Abstract:</strong><br/>
We propose isomorphism-type identities for nonlinear functionals of general infinitely divisible processes. Such identities can be viewed as an analogy of the Cameron–Martin formula for Poissonian infinitely divisible processes but with random translations. The applicability of such tools relies on precise understanding of Lévy measures of infinitely divisible processes and their representations, which are studied here in full generality. We illustrate this approach on examples of squared Bessel processes, Feller diffusions, permanental processes, as well as Lévy processes.
</p>projecteuclid.org/euclid.aop/1537862433_20180925040105Tue, 25 Sep 2018 04:01 EDTCoupling in the Heisenberg group and its applications to gradient estimateshttps://projecteuclid.org/euclid.aop/1537862434<strong>Sayan Banerjee</strong>, <strong>Maria Gordina</strong>, <strong>Phanuel Mariano</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3275--3312.</p><p><strong>Abstract:</strong><br/>
We construct a non-Markovian coupling for hypoelliptic diffusions which are Brownian motions in the three-dimensional Heisenberg group. We then derive properties of this coupling such as estimates on the coupling rate, and upper and lower bounds on the total variation distance between the laws of the Brownian motions. Finally, we use these properties to prove gradient estimates for harmonic functions for the hypoelliptic Laplacian which is the generator of Brownian motion in the Heisenberg group.
</p>projecteuclid.org/euclid.aop/1537862434_20180925040105Tue, 25 Sep 2018 04:01 EDTFirst-passage times for random walks with nonidentically distributed incrementshttps://projecteuclid.org/euclid.aop/1537862435<strong>Denis Denisov</strong>, <strong>Alexander Sakhanenko</strong>, <strong>Vitali Wachtel</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3313--3350.</p><p><strong>Abstract:</strong><br/>
We consider random walks with independent but not necessarily identical distributed increments. Assuming that the increments satisfy the well-known Lindeberg condition, we investigate the asymptotic behaviour of first-passage times over moving boundaries. Furthermore, we prove that a properly rescaled random walk conditioned to stay above the boundary up to time $n$ converges, as $n\to\infty$, towards the Brownian meander.
</p>projecteuclid.org/euclid.aop/1537862435_20180925040105Tue, 25 Sep 2018 04:01 EDTCanonical supermartingale couplingshttps://projecteuclid.org/euclid.aop/1537862436<strong>Marcel Nutz</strong>, <strong>Florian Stebegg</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3351--3398.</p><p><strong>Abstract:</strong><br/>
Two probability distributions $\mu$ and $\nu$ in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice? We construct and investigate two couplings which arise as optimizers for constrained Monge–Kantorovich optimal transport problems where only supermartingales are allowed as transports. Much like the Hoeffding–Fréchet coupling of classical transport and its symmetric counterpart, the antitone coupling, these can be characterized by order-theoretic minimality properties, as simultaneous optimal transports for certain classes of reward (or cost) functions, and through no-crossing conditions on their supports; however, our two couplings have asymmetric geometries. Remarkably, supermartingale optimal transport decomposes into classical and martingale transport in several ways.
</p>projecteuclid.org/euclid.aop/1537862436_20180925040105Tue, 25 Sep 2018 04:01 EDTA weak version of path-dependent functional Itô calculushttps://projecteuclid.org/euclid.aop/1537862437<strong>Dorival Leão</strong>, <strong>Alberto Ohashi</strong>, <strong>Alexandre B. Simas</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3399--3441.</p><p><strong>Abstract:</strong><br/>
We introduce a variational theory for processes adapted to the multidimensional Brownian motion filtration that provides a differential structure allowing to describe infinitesimal evolution of Wiener functionals at very small scales. The main novel idea is to compute the “sensitivities” of processes, namely derivatives of martingale components and a weak notion of infinitesimal generator, via a finite-dimensional approximation procedure based on controlled inter-arrival times and approximating martingales. The theory comes with convergence results that allow to interpret a large class of Wiener functionals beyond semimartingales as limiting objects of differential forms which can be computed path wisely over finite-dimensional spaces. The theory reveals that solutions of BSDEs are minimizers of energy functionals w.r.t. Brownian motion driving noise.
</p>projecteuclid.org/euclid.aop/1537862437_20180925040105Tue, 25 Sep 2018 04:01 EDTLower bounds for the smallest singular value of structured random matriceshttps://projecteuclid.org/euclid.aop/1537862438<strong>Nicholas Cook</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3442--3500.</p><p><strong>Abstract:</strong><br/>
We obtain lower tail estimates for the smallest singular value of random matrices with independent but nonidentically distributed entries. Specifically, we consider $n\times n$ matrices with complex entries of the form
\[M=A\circ X+B=(a_{ij}\xi_{ij}+b_{ij}),\] where $X=(\xi_{ij})$ has i.i.d. centered entries of unit variance and $A$ and $B$ are fixed matrices. In our main result, we obtain polynomial bounds on the smallest singular value of $M$ for the case that $A$ has bounded (possibly zero) entries, and $B=Z\sqrt{n}$ where $Z$ is a diagonal matrix with entries bounded away from zero. As a byproduct of our methods we can also handle general perturbations $B$ under additional hypotheses on $A$, which translate to connectivity hypotheses on an associated graph. In particular, we extend a result of Rudelson and Zeitouni for Gaussian matrices to allow for general entry distributions satisfying some moment hypotheses. Our proofs make use of tools which (to our knowledge) were previously unexploited in random matrix theory, in particular Szemerédi’s regularity lemma, and a version of the restricted invertibility theorem due to Spielman and Srivastava.
</p>projecteuclid.org/euclid.aop/1537862438_20180925040105Tue, 25 Sep 2018 04:01 EDTThe scaling limits of the Minimal Spanning Tree and Invasion Percolation in the planehttps://projecteuclid.org/euclid.aop/1537862439<strong>Christophe Garban</strong>, <strong>Gábor Pete</strong>, <strong>Oded Schramm</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3501--3557.</p><p><strong>Abstract:</strong><br/>
We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of the triangular lattice in the complex plane have unique scaling limits, which are invariant under rotations, scalings, and, in the case of the $\mathsf{MST}$, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting $\mathsf{MST}$. The topology of convergence is the space of spanning trees introduced by Aizenman et al. [ Random Structures Algorithms 15 (1999) 319–365], and the proof relies on the existence and conformal covariance of the scaling limit of the near-critical percolation ensemble, established in our earlier works.
</p>projecteuclid.org/euclid.aop/1537862439_20180925040105Tue, 25 Sep 2018 04:01 EDTQuenched central limit theorem for random walks in doubly stochastic random environmenthttps://projecteuclid.org/euclid.aop/1537862440<strong>Bálint Tóth</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3558--3577.</p><p><strong>Abstract:</strong><br/>
We prove the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the $H_{-1}$-condition, with slightly stronger, $\mathcal{L}^{2+\varepsilon}$ (rather than $\mathcal{L}^{2}$) integrability condition on the stream tensor. On the way we extend Nash’s moment bound to the nonreversible, divergence-free drift case, with unbounded ($\mathcal{L}^{2+\varepsilon}$) stream tensor. This paper is a sequel of [ Ann. Probab. 45 (2017) 4307–4347] and relies on technical results quoted from there.
</p>projecteuclid.org/euclid.aop/1537862440_20180925040105Tue, 25 Sep 2018 04:01 EDTThe KLS isoperimetric conjecture for generalized Orlicz ballshttps://projecteuclid.org/euclid.aop/1537862441<strong>Alexander V. Kolesnikov</strong>, <strong>Emanuel Milman</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3578--3615.</p><p><strong>Abstract:</strong><br/>
What is the optimal way to cut a convex bounded domain $K$ in Euclidean space $(\mathbb{R}^{n},|\cdot|)$ into two halves of equal volume, so that the interface between the two halves has least surface area? A conjecture of Kannan, Lovász and Simonovits asserts that, if one does not mind gaining a universal numerical factor (independent of $n$) in the surface area, one might as well dissect $K$ using a hyperplane. This conjectured essential equivalence between the former nonlinear isoperimetric inequality and its latter linear relaxation, has been shown over the last two decades to be of fundamental importance to the understanding of volume-concentration and spectral properties of convex domains. In this work, we address the conjecture for the subclass of generalized Orlicz balls
\[K=\{x\in\mathbb{R}^{n};\sum_{i=1}^{n}V_{i}(x_{i})\leq E\},\] confirming its validity for certain levels $E\in\mathbb{R}$ under a mild technical assumption on the growth of the convex functions $V_{i}$ at infinity [without which we confirm the conjecture up to a $\log(1+n)$ factor]. In sharp contrast to previous approaches for tackling the KLS conjecture, we emphasize that no symmetry is required from $K$. This significantly enlarges the subclass of convex bodies for which the conjecture is confirmed.
</p>projecteuclid.org/euclid.aop/1537862441_20180925040105Tue, 25 Sep 2018 04:01 EDTA Stratonovich–Skorohod integral formula for Gaussian rough pathshttps://projecteuclid.org/euclid.aop/1544691617<strong>Thomas Cass</strong>, <strong>Nengli Lim</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 1--60.</p><p><strong>Abstract:</strong><br/>
Given a Gaussian process $X$, its canonical geometric rough path lift $\mathbf{X}$, and a solution $Y$ to the rough differential equation (RDE) $\mathrm{d}Y_{t}=V(Y_{t})\circ\mathrm{d}\mathbf{X}_{t}$, we present a closed-form correction formula for $\int Y\circ\mathrm{d}\mathbf{X}-\int Y\,\mathrm{d}X$, that is, the difference between the rough and Skorohod integrals of $Y$ with respect to $X$. When $X$ is standard Brownian motion, we recover the classical Stratonovich-to-Itô conversion formula, which we generalize to Gaussian rough paths with finite $p$-variation, $p<3$, and satisfying an additional natural condition. This encompasses many familiar examples, including fractional Brownian motion with $H>\frac{1}{3}$. To prove the formula, we first show that the Riemann-sum approximants of the Skorohod integral converge in $L^{2}(\Omega)$ by using a novel characterization of the Cameron–Martin norm in terms of higher-dimensional Young–Stieltjes integrals. Next, we append the approximants of the Skorohod integral with a suitable compensation term without altering the limit, and the formula is finally obtained after a rebalancing of terms.
</p>projecteuclid.org/euclid.aop/1544691617_20181213040100Thu, 13 Dec 2018 04:01 ESTBerry–Esseen bounds of normal and nonnormal approximation for unbounded exchangeable pairshttps://projecteuclid.org/euclid.aop/1544691618<strong>Qi-Man Shao</strong>, <strong>Zhuo-Song Zhang</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 61--108.</p><p><strong>Abstract:</strong><br/>
An exchangeable pair approach is commonly taken in the normal and nonnormal approximation using Stein’s method. It has been successfully used to identify the limiting distribution and provide an error of approximation. However, when the difference of the exchangeable pair is not bounded by a small deterministic constant, the error bound is often not optimal. In this paper, using the exchangeable pair approach of Stein’s method, a new Berry–Esseen bound for an arbitrary random variable is established without a bound on the difference of the exchangeable pair. An optimal convergence rate for normal and nonnormal approximation is achieved when the result is applied to various examples including the quadratic forms, general Curie–Weiss model, mean field Heisenberg model and colored graph model.
</p>projecteuclid.org/euclid.aop/1544691618_20181213040100Thu, 13 Dec 2018 04:01 ESTStructure of optimal martingale transport plans in general dimensionshttps://projecteuclid.org/euclid.aop/1544691619<strong>Nassif Ghoussoub</strong>, <strong>Young-Heon Kim</strong>, <strong>Tongseok Lim</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 109--164.</p><p><strong>Abstract:</strong><br/>
Given two probability measures $\mu$ and $\nu$ in “convex order” on $\mathbb{R}^{d}$, we study the profile of one-step martingale plans $\pi$ on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ that optimize the expected value of the modulus of their increment among all martingales having $\mu$ and $\nu$ as marginals. While there is a great deal of results for the real line (i.e., when $d=1$), much less is known in the richer and more delicate higher-dimensional case that we tackle in this paper. We show that many structural results can be obtained, provided the initial measure $\mu$ is absolutely continuous with respect to the Lebesgue measure. One such a property is that $\mu$-almost every $x$ in $\mathbb{R}^{d}$ is transported by the optimal martingale plan into a probability measure $\pi_{x}$ concentrated on the extreme points of the closed convex hull of its support. This will be established for the distance cost $c(x,y)=\vert x-y\vert $ in the two-dimensional case, and also for any $d\geq3$ as long as the marginals are in “subharmonic order.” In some cases, $\pi_{x}$ is supported on the vertices of a $k(x)$-dimensional polytope, such as when the target measure is discrete. Duality plays a crucial role in our approach, even though, in contrast to standard optimal transports, the dual extremal problem may not be attained in general. We show however that “martingale supporting” Borel subsets of $\mathbb{R}^{d}\times\mathbb{R}^{d}$ can be decomposed into a collection of mutually disjoint components by means of a “convex paving” of the source space, in such a way that when the martingale is optimal for a general cost function, each of the components then supports a restricted optimal martingale transport whose dual problem is attained. This decomposition is used to obtain structural results in cases where global duality is not attained. On the other hand, it shows that certain “optimal martingale supporting” Borel sets can be viewed as higher-dimensional versions of Nikodym-type sets. The paper focuses on the distance cost, but much of the results hold for general Lipschitz cost functions.
</p>projecteuclid.org/euclid.aop/1544691619_20181213040100Thu, 13 Dec 2018 04:01 ESTRegularization by noise and flows of solutions for a stochastic heat equationhttps://projecteuclid.org/euclid.aop/1544691620<strong>Oleg Butkovsky</strong>, <strong>Leonid Mytnik</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 165--212.</p><p><strong>Abstract:</strong><br/>
Motivated by the regularization by noise phenomenon for SDEs, we prove existence and uniqueness of the flow of solutions for the non-Lipschitz stochastic heat equation
\[\frac{\partial u}{\partial t}=\frac{1}{2}\frac{\partial^{2}u}{\partial z^{2}}+b\bigl(u(t,z)\bigr)+\dot{W}(t,z),\] where $\dot{W}$ is a space-time white noise on $\mathbb{R}_{+}\times\mathbb{R}$ and $b$ is a bounded measurable function on $\mathbb{R}$. As a byproduct of our proof, we also establish the so-called path-by-path uniqueness for any initial condition in a certain class on the same set of probability one. To obtain these results, we develop a new approach that extends Davie’s method (2007) to the context of stochastic partial differential equations.
</p>projecteuclid.org/euclid.aop/1544691620_20181213040100Thu, 13 Dec 2018 04:01 ESTBrownian motion on some spaces with varying dimensionhttps://projecteuclid.org/euclid.aop/1544691621<strong>Zhen-Qing Chen</strong>, <strong>Shuwen Lou</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 213--269.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce and study Brownian motion on a class of state spaces with varying dimension. Starting with a concrete case of such state spaces that models a big square with a flag pole, we construct a Brownian motion on it and study how heat propagates on such a space. We derive sharp two-sided global estimates on its transition density function (also called heat kernel). These two-sided estimates are of Gaussian type, but the measure on the underlying state space does not satisfy volume doubling property. Parabolic Harnack inequality fails for such a process. Nevertheless, we show Hölder regularity holds for its parabolic functions. We also derive the Green function estimates for this process on bounded smooth domains. Brownian motion on some other state spaces with varying dimension are also constructed and studied in this paper.
</p>projecteuclid.org/euclid.aop/1544691621_20181213040100Thu, 13 Dec 2018 04:01 ESTRényi divergence and the central limit theoremhttps://projecteuclid.org/euclid.aop/1544691622<strong>S. G. Bobkov</strong>, <strong>G. P. Chistyakov</strong>, <strong>F. Götze</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 270--323.</p><p><strong>Abstract:</strong><br/>
We explore properties of the $\chi^{2}$ and Rényi distances to the normal law and in particular propose necessary and sufficient conditions under which these distances tend to zero in the central limit theorem (with exact rates with respect to the increasing number of summands).
</p>projecteuclid.org/euclid.aop/1544691622_20181213040100Thu, 13 Dec 2018 04:01 ESTTowards a universality picture for the relaxation to equilibrium of kinetically constrained modelshttps://projecteuclid.org/euclid.aop/1544691623<strong>Fabio Martinelli</strong>, <strong>Cristina Toninelli</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 324--361.</p><p><strong>Abstract:</strong><br/>
Recent years have seen a great deal of progress in our understanding of bootstrap percolation models, a particular class of monotone cellular automata . In the two-dimensional lattice $\mathbb{Z}^{2}$, there is now a quite satisfactory understanding of their evolution starting from a random initial condition, with a strikingly beautiful universality picture for their critical behavior. Much less is known for their nonmonotone stochastic counterpart, namely kinetically constrained models (KCM). In KCM, each vertex is resampled (independently) at rate one by tossing a $p$-coin iff it can be infected in the next step by the bootstrap model. In particular, an infection can also heal, hence the nonmonotonicity. Besides the connection with bootstrap percolation, KCM have an interest in their own as they feature some of the most striking features of the liquid/glass transition, a major and still largely open problem in condensed matter physics. In this paper, we pave the way towards proving universality results for the characteristic time scales of KCM. Our novel and general approach gives the right tools to establish a close connection between the critical scaling of characteristic time scales for KCM and the scaling of the critical length in critical bootstrap models. When applied to the Fredrickson–Andersen $k$-facilitated models in dimension $d\ge2$, among the most studied KCM, and to the Gravner–Griffeath model, our results are close to optimal.
</p>projecteuclid.org/euclid.aop/1544691623_20181213040100Thu, 13 Dec 2018 04:01 ESTThe spectral gap of dense random regular graphshttps://projecteuclid.org/euclid.aop/1544691624<strong>Konstantin Tikhomirov</strong>, <strong>Pierre Youssef</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 362--419.</p><p><strong>Abstract:</strong><br/>
For any $\alpha\in(0,1)$ and any $n^{\alpha}\leq d\leq n/2$, we show that $\lambda({\mathbf{G}})\leq C_{\alpha}\sqrt{d}$ with probability at least $1-\frac{1}{n}$, where ${\mathbf{G}}$ is the uniform random undirected $d$-regular graph on $n$ vertices, $\lambda({\mathbf{G}})$ denotes its second largest eigenvalue (in absolute value) and $C_{\alpha}$ is a constant depending only on $\alpha$. Combined with earlier results in this direction covering the case of sparse random graphs, this completely settles the problem of estimating the magnitude of $\lambda({\mathbf{G}})$, up to a multiplicative constant, for all values of $n$ and $d$, confirming a conjecture of Vu. The result is obtained as a consequence of an estimate for the second largest singular value of adjacency matrices of random directed graphs with predefined degree sequences. As the main technical tool, we prove a concentration inequality for arbitrary linear forms on the space of matrices, where the probability measure is induced by the adjacency matrix of a random directed graph with prescribed degree sequences. The proof is a nontrivial application of the Freedman inequality for martingales, combined with self-bounding and tensorization arguments. Our method bears considerable differences compared to the approach used by Broder et al. [ SIAM J. Comput. 28 (1999) 541–573] who established the upper bound for $\lambda({\mathbf{G}})$ for $d=o(\sqrt{n})$, and to the argument of Cook, Goldstein and Johnson [ Ann. Probab. 46 (2018) 72–125] who derived a concentration inequality for linear forms and estimated $\lambda({\mathbf{G}})$ in the range $d=O(n^{2/3})$ using size-biased couplings.
</p>projecteuclid.org/euclid.aop/1544691624_20181213040100Thu, 13 Dec 2018 04:01 ESTCanonical RDEs and general semimartingales as rough pathshttps://projecteuclid.org/euclid.aop/1544691625<strong>Ilya Chevyrev</strong>, <strong>Peter K. Friz</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 420--463.</p><p><strong>Abstract:</strong><br/>
In the spirit of Marcus canonical stochastic differential equations, we study a similar notion of rough differential equations (RDEs), notably dropping the assumption of continuity prevalent in the rough path literature. A new metric is exhibited in which the solution map is a continuous function of the driving rough path and a so-called path function, which directly models the effect of the jump on the system. In a second part, we show that general multidimensional semimartingales admit canonically defined rough path lifts. An extension of Lépingle’s BDG inequality to this setting is given, and in turn leads to a number of novel limit theorems for semimartingale driven differential equations, both in law and in probability, conveniently phrased a uniformly-controlled-variations (UCV) condition (Kurtz–Protter, Jakubowski–Mémin–Pagès). A number of examples illustrate the scope of our results.
</p>projecteuclid.org/euclid.aop/1544691625_20181213040100Thu, 13 Dec 2018 04:01 ESTRate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noisehttps://projecteuclid.org/euclid.aop/1544691626<strong>Aurélien Deya</strong>, <strong>Fabien Panloup</strong>, <strong>Samy Tindel</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 464--518.</p><p><strong>Abstract:</strong><br/>
We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in(1/3,1)$ and multiplicative noise component $\sigma$. When $\sigma$ is constant and for every $H\in(0,1)$, it was proved in [ Ann. Probab. 33 (2005) 703–758] that, under some mean-reverting assumptions, such a process converges to its equilibrium at a rate of order $t^{-\alpha}$ where $\alpha\in(0,1)$ (depending on $H$). In [ Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 503–538], this result has been extended to the multiplicative case when $H>1/2$. In this paper, we obtain these types of results in the rough setting $H\in(1/3,1/2)$. Once again, we retrieve the rate orders of the additive setting. Our methods also extend the multiplicative results of [ Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 503–538] by deleting the gradient assumption on the noise coefficient $\sigma$. The main theorems include some existence and uniqueness results for the invariant distribution.
</p>projecteuclid.org/euclid.aop/1544691626_20181213040100Thu, 13 Dec 2018 04:01 ESTGlobal solutions to stochastic reaction–diffusion equations with super-linear drift and multiplicative noisehttps://projecteuclid.org/euclid.aop/1544691627<strong>Robert C. Dalang</strong>, <strong>Davar Khoshnevisan</strong>, <strong>Tusheng Zhang</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 519--559.</p><p><strong>Abstract:</strong><br/>
Let $\xi (t,x)$ denote space–time white noise and consider a reaction–diffusion equation of the form \[\dot{u}(t,x)=\frac{1}{2}u"(t,x)+b\big(u(t,x)\big)+\sigma \big(u(t,x)\big)\xi (t,x),\] on $\mathbb{R}_{+}\times [0,1]$, with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists $\varepsilon >0$ such that $\vert b(z)\vert \ge \vert z\vert (\log \vert z\vert )^{1+\varepsilon }$ for all sufficiently-large values of $\vert z\vert $. When $\sigma \equiv 0$, it is well known that such PDEs frequently have nontrivial stationary solutions. By contrast, Bonder and Groisman [ Phys. D 238 (2009) 209–215] have recently shown that there is finite-time blowup when $\sigma $ is a nonzero constant. In this paper, we prove that the Bonder–Groisman condition is unimprovable by showing that the reaction–diffusion equation with noise is “typically” well posed when $\vert b(z)\vert =O(\vert z\vert \log_{+}\vert z\vert )$ as $\vert z\vert \to \infty $. We interpret the word “typically” in two essentially-different ways without altering the conclusions of our assertions.
</p>projecteuclid.org/euclid.aop/1544691627_20181213040100Thu, 13 Dec 2018 04:01 ESTSharp interface limit for stochastically perturbed mass conserving Allen–Cahn equationhttps://projecteuclid.org/euclid.aop/1544691628<strong>Tadahisa Funaki</strong>, <strong>Satoshi Yokoyama</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 1, 560--612.</p><p><strong>Abstract:</strong><br/>
This paper studies the sharp interface limit for a mass conserving Allen–Cahn equation, added an external noise and derives a stochastically perturbed mass conserving mean curvature flow in the limit. The stochastic term destroys the precise conservation law, instead the total mass changes like a Brownian motion in time. For our equation, the comparison argument does not work, so that to study the limit we adopt the asymptotic expansion method, which extends that for deterministic equations used originally in de Mottoni and Schatzman [ Interfaces Free Bound. 12 (2010) 527–549] for the nonconservative case and then in Chen et al. [ Trans. Amer. Math. Soc. 347 (1995) 1533–1589] for the conservative case. Differently from the deterministic case, each term except the leading term appearing in the expansion of the solution in a small parameter $\varepsilon$ diverges as $\varepsilon$ tends to $0$, since our equation contains the noise which converges to a white noise and the products or the powers of the white noise diverge. To derive the error estimate for our asymptotic expansion, we need to establish the Schauder estimate for a diffusion operator with coefficients determined from higher order derivatives of the noise and their powers. We show that one can choose the noise sufficiently mild in such a manner that it converges to the white noise and at the same time its diverging speed is slow enough for establishing a necessary error estimate.
</p>projecteuclid.org/euclid.aop/1544691628_20181213040100Thu, 13 Dec 2018 04:01 ESTPhase transitions in the ASEP and stochastic six-vertex modelhttps://projecteuclid.org/euclid.aop/1551171634<strong>Amol Aggarwal</strong>, <strong>Alexei Borodin</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 613--689.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider two models in the Kardar–Parisi–Zhang (KPZ) universality class, the asymmetric simple exclusion process (ASEP) and the stochastic six-vertex model. We introduce a new class of initial data (which we call shape generalized step Bernoulli initial data ) for both of these models that generalizes the step Bernoulli initial data studied in a number of recent works on the ASEP. Under this class of initial data, we analyze the current fluctuations of both the ASEP and stochastic six-vertex model and establish the existence of a phase transition along a characteristic line, across which the fluctuation exponent changes from $1/2$ to $1/3$. On the characteristic line, the current fluctuations converge to the general (rank $k$) Baik–Ben–Arous–Péché distribution for the law of the largest eigenvalue of a critically spiked covariance matrix. For $k=1$, this was established for the ASEP by Tracy and Widom; for $k>1$ (and also $k=1$, for the stochastic six-vertex model), the appearance of these distributions in both models is new.
</p>projecteuclid.org/euclid.aop/1551171634_20190226040136Tue, 26 Feb 2019 04:01 ESTLiouville first-passage percolation: Subsequential scaling limits at high temperaturehttps://projecteuclid.org/euclid.aop/1551171635<strong>Jian Ding</strong>, <strong>Alexander Dunlap</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 690--742.</p><p><strong>Abstract:</strong><br/>
Let $\{Y_{\mathfrak{B}}(x):x\in\mathfrak{B}\}$ be a discrete Gaussian free field in a two-dimensional box $\mathfrak{B}$ of side length $S$ with Dirichlet boundary conditions. We study Liouville first-passage percolation: the shortest-path metric in which each vertex $x$ is given a weight of $e^{\gamma Y_{\mathfrak{B}}(x)}$ for some $\gamma>0$. We show that for sufficiently small but fixed $\gamma>0$, for any sequence of scales $\{S_{k}\}$ there exists a subsequence along which the appropriately scaled and interpolated Liouville FPP metric converges in the Gromov–Hausdorff sense to a random metric on the unit square in $\mathbf{R}^{2}$. In addition, all possible (conjecturally unique) scaling limits are homeomorphic by bi-Hölder-continuous homeomorphisms to the unit square with the Euclidean metric.
</p>projecteuclid.org/euclid.aop/1551171635_20190226040136Tue, 26 Feb 2019 04:01 ESTDerivative and divergence formulae for diffusion semigroupshttps://projecteuclid.org/euclid.aop/1551171636<strong>Anton Thalmaier</strong>, <strong>James Thompson</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 743--773.</p><p><strong>Abstract:</strong><br/>
For a semigroup $P_{t}$ generated by an elliptic operator on a smooth manifold $M$, we use straightforward martingale arguments to derive probabilistic formulae for $P_{t}(V(f))$, not involving derivatives of $f$, where $V$ is a vector field on $M$. For nonsymmetric generators, such formulae correspond to the derivative of the heat kernel in the forward variable. As an application, these formulae can be used to derive various shift-Harnack inequalities.
</p>projecteuclid.org/euclid.aop/1551171636_20190226040136Tue, 26 Feb 2019 04:01 ESTLow-dimensional lonely branching random walks die outhttps://projecteuclid.org/euclid.aop/1551171637<strong>Matthias Birkner</strong>, <strong>Rongfeng Sun</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 774--803.</p><p><strong>Abstract:</strong><br/>
The lonely branching random walks on $\mathbb{Z}^{d}$ is an interacting particle system where each particle moves as an independent random walk and undergoes critical binary branching when it is alone. We show that if the symmetrized walk is recurrent, lonely branching random walks die out locally. Furthermore, the same result holds if additional branching is allowed when the walk is not alone.
</p>projecteuclid.org/euclid.aop/1551171637_20190226040136Tue, 26 Feb 2019 04:01 ESTBoundary regularity of stochastic PDEshttps://projecteuclid.org/euclid.aop/1551171638<strong>Máté Gerencsér</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 804--834.</p><p><strong>Abstract:</strong><br/>
The boundary behaviour of solutions of stochastic PDEs with Dirichlet boundary conditions can be surprisingly—and in a sense, arbitrarily—bad: as shown by Krylov [ SIAM J. Math. Anal. 34 (2003) 1167–1182], for any $\alpha>0$ one can find a simple $1$-dimensional constant coefficient linear equation whose solution at the boundary is not $\alpha$-Hölder continuous.
We obtain a positive counterpart of this: under some mild regularity assumptions on the coefficients, solutions of semilinear SPDEs on $\mathcal{C}^{1}$ domains are proved to be $\alpha$-Hölder continuous up to the boundary with some $\alpha>0$.
</p>projecteuclid.org/euclid.aop/1551171638_20190226040136Tue, 26 Feb 2019 04:01 ESTLimit theory for geometric statistics of point processes having fast decay of correlationshttps://projecteuclid.org/euclid.aop/1551171639<strong>B. Błaszczyszyn</strong>, <strong>D. Yogeshwaran</strong>, <strong>J. E. Yukich</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 835--895.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{P}$ be a simple, stationary point process on $\mathbb{R}^{d}$ having fast decay of correlations, that is, its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let $\mathcal{P}_{n}:=\mathcal{P}\cap W_{n}$ be its restriction to windows $W_{n}:=[-{\frac{1}{2}}n^{1/d},{\frac{1}{2}}n^{1/d}]^{d}\subset\mathbb{R}^{d}$. We consider the statistic $H_{n}^{\xi}:=\sum_{x\in\mathcal{P}_{n}}\xi(x,\mathcal{P}_{n})$ where $\xi(x,\mathcal{P}_{n})$ denotes a score function representing the interaction of $x$ with respect to $\mathcal{P}_{n}$. When $\xi$ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics and central limit theorems for $H_{n}^{\xi}$ and, more generally, for statistics of the re-scaled, possibly signed, $\xi$-weighted point measures $\mu_{n}^{\xi}:=\sum_{x\in\mathcal{P}_{n}}\xi(x,\mathcal{P}_{n})\delta_{n^{-1/d}x}$, as $W_{n}\uparrow\mathbb{R}^{d}$. This gives the limit theory for nonlinear geometric statistics (such as clique counts, the number of Morse critical points, intrinsic volumes of the Boolean model and total edge length of the $k$-nearest neighbors graph) of $\alpha$-determinantal point processes (for $-1/\alpha\in\mathbb{N}$) having fast decreasing kernels, including the $\beta$-Ginibre ensembles, extending the Gaussian fluctuation results of Soshnikov [ Ann. Probab. 30 (2002) 171–187] to nonlinear statistics. It also gives the limit theory for geometric $U$-statistics of $\alpha$-permanental point processes (for $1/\alpha\in\mathbb{N}$) as well as the zero set of Gaussian entire functions, extending the central limit theorems of Nazarov and Sodin [ Comm. Math. Phys. 310 (2012) 75–98] and Shirai and Takahashi [ J. Funct. Anal. 205 (2003) 414–463], which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in [ Stochastic Process. Appl. 56 (1995) 321–335; Statist. Probab. Lett. 36 (1997) 299–306] to show the fast decay of the correlations of $\xi$-weighted point measures. The latter property is shown to imply a condition equivalent to Brillinger mixing, and consequently yields the asymptotic normality of $\mu_{n}^{\xi}$ via an extension of the cumulant method.
</p>projecteuclid.org/euclid.aop/1551171639_20190226040136Tue, 26 Feb 2019 04:01 ESTDifferential subordination under change of lawhttps://projecteuclid.org/euclid.aop/1551171640<strong>Komla Domelevo</strong>, <strong>Stefanie Petermichl</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 896--925.</p><p><strong>Abstract:</strong><br/>
We prove optimal $L^{2}$ bounds for a pair of Hilbert space valued differentially subordinate martingales under a change of law. The change of law is given by a process called a weight and sharpness, and in this context refers to the optimal growth with respect to the characteristic of the weight. The pair of martingales is adapted, uniformly integrable and càdlàg. Differential subordination is in the sense of Burkholder, defined through the use of the square bracket. In the scalar dyadic setting with underlying Lebesgue measure, this was proved by Wittwer [ Math. Res. Lett. 7 (2000) 1–12], where homogeneity was heavily used. Recent progress by Thiele–Treil–Volberg [ Adv. Math. 285 (2015) 1155–1188] and Lacey [ Israel J. Math. 217 (2017) 181–195] independently resolved the so-called nonhomogenous case using discrete in time filtrations, where one martingale is a predictable multiplier of the other. The general case for continuous-in-time filtrations and pairs of martingales that are not necessarily predictable multipliers, remained open and is addressed here. As a very useful second main result, we give the explicit expression of a Bellman function of four variables for the weighted estimate of subordinate martingales with jumps. This construction includes an analysis of the regularity of this function as well as a very precise convexity, needed to deal with the jump part.
</p>projecteuclid.org/euclid.aop/1551171640_20190226040136Tue, 26 Feb 2019 04:01 ESTCentral limit theorems for empirical transportation cost in general dimensionhttps://projecteuclid.org/euclid.aop/1551171641<strong>Eustasio del Barrio</strong>, <strong>Jean-Michel Loubes</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 926--951.</p><p><strong>Abstract:</strong><br/>
We consider the problem of optimal transportation with quadratic cost between a empirical measure and a general target probability on $\mathbb{R}^{d}$, with $d\geq1$. We provide new results on the uniqueness and stability of the associated optimal transportation potentials, namely, the minimizers in the dual formulation of the optimal transportation problem. As a consequence, we show that a CLT holds for the empirical transportation cost under mild moment and smoothness requirements. The limiting distributions are Gaussian and admit a simple description in terms of the optimal transportation potentials.
</p>projecteuclid.org/euclid.aop/1551171641_20190226040136Tue, 26 Feb 2019 04:01 ESTDeterminantal spanning forests on planar graphshttps://projecteuclid.org/euclid.aop/1551171642<strong>Richard Kenyon</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 952--988.</p><p><strong>Abstract:</strong><br/>
We generalize the uniform spanning tree to construct a family of determinantal measures on essential spanning forests on periodic planar graphs in which every component tree is bi-infinite. Like the uniform spanning tree, these measures arise naturally from the Laplacian on the graph.
More generally, these results hold for the “massive” Laplacian determinant which counts rooted spanning forests with weight $M$ per finite component. These measures typically have a form of conformal invariance, unlike the usual rooted spanning tree measure. We show that the spectral curve for these models is always a simple Harnack curve; this fact controls the decay of edge-edge correlations in these models.
We construct a limit shape theory in these settings, where the limit shapes are defined by measured foliations of fixed isotopy type.
</p>projecteuclid.org/euclid.aop/1551171642_20190226040136Tue, 26 Feb 2019 04:01 ESTComparison principle for stochastic heat equation on $\mathbb{R}^{d}$https://projecteuclid.org/euclid.aop/1551171643<strong>Le Chen</strong>, <strong>Jingyu Huang</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 989--1035.</p><p><strong>Abstract:</strong><br/>
We establish the strong comparison principle and strict positivity of solutions to the following nonlinear stochastic heat equation on $\mathbb{R}^{d}$
\[\biggl(\frac{\partial}{\partial t}-\frac{1}{2}\Delta \biggr)u(t,x)=\rho\bigl(u(t,x)\bigr)\dot{M}(t,x),\] for measure-valued initial data, where $\dot{M}$ is a spatially homogeneous Gaussian noise that is white in time and $\rho$ is Lipschitz continuous. These results are obtained under the condition that $\int_{\mathbb{R}^{d}}(1+|\xi|^{2})^{\alpha-1}\hat{f}(\text{d}\xi)<\infty$ for some $\alpha\in(0,1]$, where $\hat{f}$ is the spectral measure of the noise. The weak comparison principle and nonnegativity of solutions to the same equation are obtained under Dalang’s condition, that is, $\alpha=0$. As some intermediate results, we obtain handy upper bounds for $L^{p}(\Omega)$-moments of $u(t,x)$ for all $p\ge2$, and also prove that $u$ is a.s. Hölder continuous with order $\alpha-\varepsilon$ in space and $\alpha/2-\varepsilon$ in time for any small $\varepsilon>0$.
</p>projecteuclid.org/euclid.aop/1551171643_20190226040136Tue, 26 Feb 2019 04:01 ESTKirillov–Frenkel character formula for loop groups, radial part and Brownian sheethttps://projecteuclid.org/euclid.aop/1551171644<strong>Manon Defosseux</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 1036--1055.</p><p><strong>Abstract:</strong><br/>
We consider the coadjoint action of a Loop group of a compact group on the dual of the corresponding centrally extended Loop algebra and prove that a Brownian motion in a Cartan subalgebra conditioned to remain in an affine Weyl chamber—which can be seen as a space time conditioned Brownian motion—is distributed as the radial part process of a Brownian sheet on the underlying Lie algebra.
</p>projecteuclid.org/euclid.aop/1551171644_20190226040136Tue, 26 Feb 2019 04:01 ESTHeat kernel upper bounds for interacting particle systemshttps://projecteuclid.org/euclid.aop/1551171645<strong>Arianna Giunti</strong>, <strong>Yu Gu</strong>, <strong>Jean-Christophe Mourrat</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 1056--1095.</p><p><strong>Abstract:</strong><br/>
We show a diffusive upper bound on the transition probability of a tagged particle in the symmetric simple exclusion process. The proof relies on optimal spectral gap estimates for the dynamics in finite volume, which are of independent interest. We also show off-diagonal estimates of Carne–Varopoulos type.
</p>projecteuclid.org/euclid.aop/1551171645_20190226040136Tue, 26 Feb 2019 04:01 ESTParacontrolled quasilinear SPDEshttps://projecteuclid.org/euclid.aop/1551171646<strong>Marco Furlan</strong>, <strong>Massimiliano Gubinelli</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 1096--1135.</p><p><strong>Abstract:</strong><br/>
We introduce a nonlinear paracontrolled calculus and use it to renormalise a class of singular SPDEs including certain quasilinear variants of the periodic two-dimensional parabolic Anderson model.
</p>projecteuclid.org/euclid.aop/1551171646_20190226040136Tue, 26 Feb 2019 04:01 ESTErdős–Feller–Kolmogorov–Petrowsky law of the iterated logarithm for self-normalized martingales: A game-theoretic approachhttps://projecteuclid.org/euclid.aop/1551171647<strong>Takeyuki Sasai</strong>, <strong>Kenshi Miyabe</strong>, <strong>Akimichi Takemura</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 1136--1161.</p><p><strong>Abstract:</strong><br/>
We prove an Erdős–Feller–Kolmogorov–Petrowsky law of the iterated logarithm for self-normalized martingales. Our proof is given in the framework of the game-theoretic probability of Shafer and Vovk. Like many other game-theoretic proofs, our proof is self-contained and explicit.
</p>projecteuclid.org/euclid.aop/1551171647_20190226040136Tue, 26 Feb 2019 04:01 ESTCritical radius and supremum of random spherical harmonicshttps://projecteuclid.org/euclid.aop/1551171648<strong>Renjie Feng</strong>, <strong>Robert J. Adler</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 1162--1184.</p><p><strong>Abstract:</strong><br/>
We first consider deterministic immersions of the $d$-dimensional sphere into high dimensional Euclidean spaces, where the immersion is via spherical harmonics of level $n$. The main result of the article is the, a priori unexpected, fact that there is a uniform lower bound to the critical radius of the immersions as $n\to\infty$. This fact has immediate implications for random spherical harmonics with fixed $L^{2}$-norm. In particular, it leads to an exact and explicit formulae for the tail probability of their (large deviation) suprema by the tube formula, and also relates this to the expected Euler characteristic of their upper level sets.
</p>projecteuclid.org/euclid.aop/1551171648_20190226040136Tue, 26 Feb 2019 04:01 ESTComponent sizes for large quantum Erdős–Rényi graph near criticalityhttps://projecteuclid.org/euclid.aop/1551171650<strong>Amir Dembo</strong>, <strong>Anna Levit</strong>, <strong>Sreekar Vadlamani</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 47, Number 2, 1185--1219.</p><p><strong>Abstract:</strong><br/>
The $N$ vertices of a quantum random graph are each a circle independently punctured at Poisson points of arrivals, with parallel connections derived through for each pair of these punctured circles by yet another independent Poisson process. Considering these graphs at their critical parameters, we show that the joint law of the rescaled by $N^{2/3}$ and ordered sizes of their connected components, converges to that of the ordered lengths of excursions above zero for a reflected Brownian motion with drift. Thereby, this work forms the first example of an inhomogeneous random graph, beyond the case of effectively rank-1 models, which is rigorously shown to be in the Erdős–Rényi graphs universality class in terms of Aldous’s results.
</p>projecteuclid.org/euclid.aop/1551171650_20190226040136Tue, 26 Feb 2019 04:01 EST