Andersen dynamics is a standard method for molecular simulations, and a precursor of the Hamiltonian Monte Carlo algorithm used in MCMC inference. The stochastic process corresponding to Andersen dynamics is a PDMP (piecewise deterministic Markov process) that iterates between Hamiltonian flows and velocity randomizations of randomly selected particles. Both from the viewpoint of molecular dynamics and MCMC inference, a basic question is to understand the convergence to equilibrium of this PDMP particularly in high dimension. Here we present couplings to obtain sharp convergence bounds in the Wasserstein sense that do not require global convexity of the underlying potential energy.
{"title":"Couplings for Andersen dynamics","authors":"Nawaf Bou-Rabee, A. Eberle","doi":"10.1214/21-AIHP1197","DOIUrl":"https://doi.org/10.1214/21-AIHP1197","url":null,"abstract":"Andersen dynamics is a standard method for molecular simulations, and a precursor of the Hamiltonian Monte Carlo algorithm used in MCMC inference. The stochastic process corresponding to Andersen dynamics is a PDMP (piecewise deterministic Markov process) that iterates between Hamiltonian flows and velocity randomizations of randomly selected particles. Both from the viewpoint of molecular dynamics and MCMC inference, a basic question is to understand the convergence to equilibrium of this PDMP particularly in high dimension. Here we present couplings to obtain sharp convergence bounds in the Wasserstein sense that do not require global convexity of the underlying potential energy.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"6 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80763134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider critical site percolation on $mathbb{Z}^d$ with $d geq 2$. Cerf (2015) pointed out that from classical work by Aizenman, Kesten and Newman (1987) and Gandolfi, Grimmett and Russo (1988) one can obtain that the two-arms exponent is at least $1/2$. The paper by Cerf slightly improves that lower bound. Except for $d=2$ and for high $d$, no upper bound for this exponent seems to be known in the literature so far (not even implicity). We show that the distance-$n$ two-arms probability is at least $c n^{-(d^2 + 4 d -2)}$ (with $c >0$ a constant which depends on $d$), thus giving an upper bound $d^2 + 4 d -2$ for the above mentioned exponent.
用$d geq 2$考虑$mathbb{Z}^d$上的关键站点渗透。Cerf(2015)指出，从Aizenman, Kesten and Newman(1987)和Gandolfi, Grimmett and Russo(1988)的经典作品中可以得到双臂指数至少为$1/2$。Cerf的论文稍微改进了这个下界。除了$d=2$和高的$d$，到目前为止，在文献中似乎没有这个指数的上界(甚至没有隐含)。我们表明，距离- $n$双臂概率至少为$c n^{-(d^2 + 4 d -2)}$ ($c >0$是一个取决于$d$的常数)，从而给出上述指数的上界$d^2 + 4 d -2$。
{"title":"An upper bound on the two-arms exponent for critical percolation on Zd","authors":"J. Berg, Diederik van Engelenburg","doi":"10.1214/21-aihp1153","DOIUrl":"https://doi.org/10.1214/21-aihp1153","url":null,"abstract":"Consider critical site percolation on $mathbb{Z}^d$ with $d geq 2$. Cerf (2015) pointed out that from classical work by Aizenman, Kesten and Newman (1987) and Gandolfi, Grimmett and Russo (1988) one can obtain that the two-arms exponent is at least $1/2$. The paper by Cerf slightly improves that lower bound. \u0000Except for $d=2$ and for high $d$, no upper bound for this exponent seems to be known in the literature so far (not even implicity). We show that the distance-$n$ two-arms probability is at least $c n^{-(d^2 + 4 d -2)}$ (with $c >0$ a constant which depends on $d$), thus giving an upper bound $d^2 + 4 d -2$ for the above mentioned exponent.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"7 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85685333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a zero-one law for the stationary measure for algebraic sets generalizing the results of Furstenberg [13] and Guivarc'h and Le Page [20]. As an application, we establish a local limit theorem for the coefficients of random walks on the general linear group.
推广了Furstenberg[13]和Guivarc'h and Le Page[20]的结果，证明了代数集平稳测度的一个0 - 1定律。作为应用，我们建立了一般线性群上随机游走系数的局部极限定理。
{"title":"A zero-one law for invariant measures and a local limit theorem for coefficients of random walks on the general linear group","authors":"I. Grama, Jean-François Quint, Hui Xiao","doi":"10.1214/21-aihp1221","DOIUrl":"https://doi.org/10.1214/21-aihp1221","url":null,"abstract":"We prove a zero-one law for the stationary measure for algebraic sets generalizing the results of Furstenberg [13] and Guivarc'h and Le Page [20]. As an application, we establish a local limit theorem for the coefficients of random walks on the general linear group.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"4 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81579743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the parabolic Anderson model (PAM) $partial_t u = frac12 Delta u + xi u$ in $mathbb R^2$ with a Gaussian (space) white-noise potential $xi$. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time $t$, written $U(t)$, is given by $log U(t)sim chi t log t$, with the deterministic constant $chi$ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour principal Dirichlet of the eigenvalue $boldsymbol lambda_1(Q_t)$ of the Anderson operator on the box $Q_t= [-frac{t}{2},frac{t}{2}]^2$ by $boldsymbol lambda_1(Q_t)simchilog t$.
我们考虑抛物线型安德森模型(PAM)。 $partial_t u = frac12 Delta u + xi u$ 在 $mathbb R^2$ 具有高斯(空间)白噪声势 $xi$． 我们证明了总质量在时间上的几乎确定的大时渐近行为 $t$，写的 $U(t)$，由 $log U(t)sim chi t log t$，带有确定性常数 $chi$ 用变分公式表示。在一位作者的早期工作中，这个常数被用来描述特征值的渐近行为主狄利克雷 $boldsymbol lambda_1(Q_t)$ 安德森接线员的电话 $Q_t= [-frac{t}{2},frac{t}{2}]^2$ 通过 $boldsymbol lambda_1(Q_t)simchilog t$．
{"title":"Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential","authors":"W. Konig, Nicolas Perkowski, W. V. Zuijlen","doi":"10.1214/21-aihp1215","DOIUrl":"https://doi.org/10.1214/21-aihp1215","url":null,"abstract":"We consider the parabolic Anderson model (PAM) $partial_t u = frac12 Delta u + xi u$ in $mathbb R^2$ with a Gaussian (space) white-noise potential $xi$. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time $t$, written $U(t)$, is given by $log U(t)sim chi t log t$, with the deterministic constant $chi$ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour principal Dirichlet of the eigenvalue $boldsymbol lambda_1(Q_t)$ of the Anderson operator on the box $Q_t= [-frac{t}{2},frac{t}{2}]^2$ by $boldsymbol lambda_1(Q_t)simchilog t$.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"28 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75610413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The set of finite binary matrices of a given size is known to carry a finite type A bicrystal structure. We first review this classical construction, explain how it yields a short proof of the equality between Kostka polynomials and one-dimensional sums together with a natural generalisation of the 2M − X Pitman transform. Next, we show that, once the relevant formalism on families of infinite binary matrices is introduced, this is a particular case of a much more general phenomenon. Each such family of matrices is proved to be endowed with Kac-Moody bicrystal and tricrystal structures defined from the classical root systems. Moreover, we give an explicit decomposition of these multicrystals, reminiscent of the decomposition of characters yielding the Cauchy identities.
{"title":"Duality and bicrystals on infinite binary matrices","authors":"Thomas Gerber, C. Lecouvey","doi":"10.4171/aihpd/165","DOIUrl":"https://doi.org/10.4171/aihpd/165","url":null,"abstract":"The set of finite binary matrices of a given size is known to carry a finite type A bicrystal structure. We first review this classical construction, explain how it yields a short proof of the equality between Kostka polynomials and one-dimensional sums together with a natural generalisation of the 2M − X Pitman transform. Next, we show that, once the relevant formalism on families of infinite binary matrices is introduced, this is a particular case of a much more general phenomenon. Each such family of matrices is proved to be endowed with Kac-Moody bicrystal and tricrystal structures defined from the classical root systems. Moreover, we give an explicit decomposition of these multicrystals, reminiscent of the decomposition of characters yielding the Cauchy identities.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"21 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89038426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider random walks on the support of a random purely atomic measure on $mathbb{R}^d$ with random jump probability rates. The jump range can be unbounded. The purely atomic measure is reversible for the random walk and stationary for the action of the group $mathbb{G}=mathbb{R}^d$ or $mathbb{G}=mathbb{Z}^d$. By combining two-scale convergence and Palm theory for $mathbb{G}$-stationary random measures and by developing a cut-off procedure, under suitable second moment conditions we prove for almost all environments the homogenization for the massive Poisson equation of the associated Markov generators. In addition, we obtain the quenched convergence of the $L^2$-Markov semigroup and resolvent of the diffusively rescaled random walk to the corresponding ones of the Brownian motion with covariance matrix $2D$. For symmetric jump rates, the above convergence plays a crucial role in the derivation of hydrodynamic limits when considering multiple random walks with site-exclusion or zero range interaction. We do not require any ellipticity assumption, neither non-degeneracy of the homogenized matrix $D$. Our results cover a large family of models, including e.g. random conductance models on $mathbb{Z}^d$ and on general lattices (possibly with long conductances), Mott variable range hopping, simple random walks on Delaunay triangulations, simple random walks on supercritical percolation clusters.
{"title":"Stochastic homogenization of random walks on point processes","authors":"A. Faggionato","doi":"10.1214/22-aihp1269","DOIUrl":"https://doi.org/10.1214/22-aihp1269","url":null,"abstract":"We consider random walks on the support of a random purely atomic measure on $mathbb{R}^d$ with random jump probability rates. The jump range can be unbounded. The purely atomic measure is reversible for the random walk and stationary for the action of the group $mathbb{G}=mathbb{R}^d$ or $mathbb{G}=mathbb{Z}^d$. By combining two-scale convergence and Palm theory for $mathbb{G}$-stationary random measures and by developing a cut-off procedure, under suitable second moment conditions we prove for almost all environments the homogenization for the massive Poisson equation of the associated Markov generators. In addition, we obtain the quenched convergence of the $L^2$-Markov semigroup and resolvent of the diffusively rescaled random walk to the corresponding ones of the Brownian motion with covariance matrix $2D$. For symmetric jump rates, the above convergence plays a crucial role in the derivation of hydrodynamic limits when considering multiple random walks with site-exclusion or zero range interaction. We do not require any ellipticity assumption, neither non-degeneracy of the homogenized matrix $D$. Our results cover a large family of models, including e.g. random conductance models on $mathbb{Z}^d$ and on general lattices (possibly with long conductances), Mott variable range hopping, simple random walks on Delaunay triangulations, simple random walks on supercritical percolation clusters.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"37 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81086418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the distribution of the supremum of the Airy process with $m$ wanderers minus a parabola, or equivalently the limit of the rescaled maximal height of a system of $N$ non-intersecting Brownian bridges as $Ntoinfty$, where the first $N-m$ paths start and end at the origin and the remaining $m$ go between arbitrary positions. The distribution provides a $2m$-parameter deformation of the Tracy--Widom GOE distribution, which is recovered in the limit corresponding to all Brownian paths starting and ending at the origin. We provide several descriptions of this distribution function: (i) A Fredholm determinant formula; (ii) A formula in terms of Painleve II functions; (iii) A representation as a marginal of the KPZ fixed point with initial data given as the top path in a stationary system of reflected Brownian motions with drift; (iv) A characterization as the solution of a version of the Bloemendal--Virag PDE (arXiv:1011.1877, arXiv:1109.3704) for spiked Tracy--Widom distributions; (v) A representation as a solution of the KdV equation. We also discuss connections with a model of last passage percolation with boundary sources.
{"title":"Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy–Widom GOE distribution","authors":"Karl Liechty, G. Nguyen, Daniel Remenik","doi":"10.1214/21-AIHP1229","DOIUrl":"https://doi.org/10.1214/21-AIHP1229","url":null,"abstract":"We study the distribution of the supremum of the Airy process with $m$ wanderers minus a parabola, or equivalently the limit of the rescaled maximal height of a system of $N$ non-intersecting Brownian bridges as $Ntoinfty$, where the first $N-m$ paths start and end at the origin and the remaining $m$ go between arbitrary positions. The distribution provides a $2m$-parameter deformation of the Tracy--Widom GOE distribution, which is recovered in the limit corresponding to all Brownian paths starting and ending at the origin. \u0000We provide several descriptions of this distribution function: (i) A Fredholm determinant formula; (ii) A formula in terms of Painleve II functions; (iii) A representation as a marginal of the KPZ fixed point with initial data given as the top path in a stationary system of reflected Brownian motions with drift; (iv) A characterization as the solution of a version of the Bloemendal--Virag PDE (arXiv:1011.1877, arXiv:1109.3704) for spiked Tracy--Widom distributions; (v) A representation as a solution of the KdV equation. We also discuss connections with a model of last passage percolation with boundary sources.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"56 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81371283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We consider causal 3-dimensional triangulations with the topology of S 2 × [0 , 1] or D 2 × [0 , 1] where S 2 and D 2 are the 2-dimensional sphere and disc, respectively. These triangulations consist of slices and we show that these slices can be mapped bijectively onto a set of certain coloured 2-dimensional cell complexes satisfying simple conditions. The cell complexes arise as the cross section of the individual slices.
{"title":"The structure of spatial slices of 3-dimensional causal triangulations","authors":"B. Durhuus, T. Jonsson","doi":"10.4171/aihpd/91","DOIUrl":"https://doi.org/10.4171/aihpd/91","url":null,"abstract":". We consider causal 3-dimensional triangulations with the topology of S 2 × [0 , 1] or D 2 × [0 , 1] where S 2 and D 2 are the 2-dimensional sphere and disc, respectively. These triangulations consist of slices and we show that these slices can be mapped bijectively onto a set of certain coloured 2-dimensional cell complexes satisfying simple conditions. The cell complexes arise as the cross section of the individual slices.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"54 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91377185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Using some basic notions from the theory of Hopf algebras and quasi-shuffle algebras, we introduce rigorously a new family of rough paths: the quasi-geometric rough paths. We discuss their main properties. In particular, we will relate them with iterated Brownian integrals and the concept of "simple bracket extension", developed in the PhD thesis of David Kelly. As a consequence of these results, we have a sufficient criterion to show for any $gammain (0,1)$ and any sufficiently smooth function $varphi colon mathbb{R}^dto mathbb{R}$ a rough change of variable formula on any $gamma$-Holder continuous path $xcolon [0, T]to mathbb{R}^d$, i.e. an explicit expression of $varphi(x_t)$ in terms of rough integrals.
{"title":"Quasi-geometric rough paths and rough change of variable formula","authors":"C. Bellingeri","doi":"10.1214/22-aihp1297","DOIUrl":"https://doi.org/10.1214/22-aihp1297","url":null,"abstract":"Using some basic notions from the theory of Hopf algebras and quasi-shuffle algebras, we introduce rigorously a new family of rough paths: the quasi-geometric rough paths. We discuss their main properties. In particular, we will relate them with iterated Brownian integrals and the concept of \"simple bracket extension\", developed in the PhD thesis of David Kelly. As a consequence of these results, we have a sufficient criterion to show for any $gammain (0,1)$ and any sufficiently smooth function $varphi colon mathbb{R}^dto mathbb{R}$ a rough change of variable formula on any $gamma$-Holder continuous path $xcolon [0, T]to mathbb{R}^d$, i.e. an explicit expression of $varphi(x_t)$ in terms of rough integrals.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"47 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86614538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Berestycki, E. Brunet, A. Cortines, Bastien Mallein
In this article, we study the extremal processes of branching Brownian motions conditioned on having an unusually large maximum. The limiting point measures form a one-parameter family and are the decoration point measures in the extremal processes of several branching processes, including branching Brownian motions with variable speed and multitype branching Brownian motions. We give a new, alternative representation of these point measures and we show that they form a continuous family. This also yields a simple probabilistic expression for the constant that appears in the large deviation probability of having a large displacement. As an application, we show that Bovier and Hartung (2015)'s results about variable speed branching Brownian motion also describe the extremal point process of branching Ornstein-Uhlenbeck processes.
{"title":"A simple backward construction of branching Brownian motion with large displacement and applications","authors":"J. Berestycki, E. Brunet, A. Cortines, Bastien Mallein","doi":"10.1214/21-aihp1212","DOIUrl":"https://doi.org/10.1214/21-aihp1212","url":null,"abstract":"In this article, we study the extremal processes of branching Brownian motions conditioned on having an unusually large maximum. The limiting point measures form a one-parameter family and are the decoration point measures in the extremal processes of several branching processes, including branching Brownian motions with variable speed and multitype branching Brownian motions. We give a new, alternative representation of these point measures and we show that they form a continuous family. This also yields a simple probabilistic expression for the constant that appears in the large deviation probability of having a large displacement. As an application, we show that Bovier and Hartung (2015)'s results about variable speed branching Brownian motion also describe the extremal point process of branching Ornstein-Uhlenbeck processes.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"51 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86147811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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