We prove that the set of possible values for the percolation threshold $p_c$ of Cayley graphs has a gap at 1 in the sense that there exists $varepsilon_0>0$ such that for every Cayley graph $G$ one either has $p_c(G)=1$ or $p_c(G) leq 1-varepsilon_0$. The proof builds on the new approach of Duminil-Copin, Goswami, Raoufi, Severo&Yadin to the existence of phase transition using the Gaussian free field, combined with the finitary version of Gromov's theorem on the structure of groups of polynomial growth of Breuillard, Green&Tao.
{"title":"Gap at 1 for the percolation threshold of Cayley graphs","authors":"C. Panagiotis, Franco Severo","doi":"10.1214/22-aihp1286","DOIUrl":"https://doi.org/10.1214/22-aihp1286","url":null,"abstract":"We prove that the set of possible values for the percolation threshold $p_c$ of Cayley graphs has a gap at 1 in the sense that there exists $varepsilon_0>0$ such that for every Cayley graph $G$ one either has $p_c(G)=1$ or $p_c(G) leq 1-varepsilon_0$. The proof builds on the new approach of Duminil-Copin, Goswami, Raoufi, Severo&Yadin to the existence of phase transition using the Gaussian free field, combined with the finitary version of Gromov's theorem on the structure of groups of polynomial growth of Breuillard, Green&Tao.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"1 5","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72630050","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 investigate a degree preserving variant of the $Delta$-Y transformation which replaces a triangle with a new 6-valent vertex which has double edges to the vertices that had been in the triangle. This operation is relevant for understanding scalar Feynman integrals in 6 dimensions. We study the structure of equivalence classes under this operation and its inverse, with particular attention to when the equivalence classes are finite, when they contain simple 6-regular graphs, and when they contain doubled 3-regular graphs. The last of these, in particular, is relevant for the Feynman integral calculations and we make some observations linking the structure of these classes to the Feynman periods. Furthermore, we investigate properties of minimal graphs in these equivalence classes.
{"title":"A degree preserving delta wye transformation with applications to 6-regular graphs and Feynman periods","authors":"S. Jeffries, K. Yeats","doi":"10.4171/aihpd/172","DOIUrl":"https://doi.org/10.4171/aihpd/172","url":null,"abstract":"We investigate a degree preserving variant of the $Delta$-Y transformation which replaces a triangle with a new 6-valent vertex which has double edges to the vertices that had been in the triangle. This operation is relevant for understanding scalar Feynman integrals in 6 dimensions. We study the structure of equivalence classes under this operation and its inverse, with particular attention to when the equivalence classes are finite, when they contain simple 6-regular graphs, and when they contain doubled 3-regular graphs. The last of these, in particular, is relevant for the Feynman integral calculations and we make some observations linking the structure of these classes to the Feynman periods. Furthermore, we investigate properties of minimal graphs in these equivalence classes.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"98 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77959468","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 give a procedure to construct (quasi-)trisection diagrams for closed (pseudo-)manifolds generated by colored tensor models without restrictions on the number of simplices in the triangulation, therefore generalizing previous works in the context of crystallizations and PL-manifolds. We further speculate on generalization of similar constructions for a class of pseudo-manifolds generated by simplicial colored tensor models.
{"title":"Trisections in colored tensor models","authors":"Riccardo Martini, R. Toriumi","doi":"10.4171/aihpd/167","DOIUrl":"https://doi.org/10.4171/aihpd/167","url":null,"abstract":"We give a procedure to construct (quasi-)trisection diagrams for closed (pseudo-)manifolds generated by colored tensor models without restrictions on the number of simplices in the triangulation, therefore generalizing previous works in the context of crystallizations and PL-manifolds. We further speculate on generalization of similar constructions for a class of pseudo-manifolds generated by simplicial colored tensor models.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"13 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77321181","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 a stochastic heat equation of the type, $partial_t u = partial^2_x u + sigma(u)dot{W}$ on $(0,,infty)times[-1,,1]$ with periodic boundary conditions and on-degenerate positive initial data, where $sigma:mathbb{R} tomathbb{R}$ is a non-random Lipschitz continuous function and $dot{W}$ denotes space-time white noise. If additionally $sigma(0)=0$ then the solution is known to be strictly positive; see Mueller '91. In that case, we prove that the oscillation of the logarithm of the solution decays sublinearly as time tends to infinity. Among other things, it follows that, with probability one, all limit points of $t^{-1}, sup_{xin[-1,1]}, log u(t,,x)$ and $t^{-1}, inf_{xin[-1,1]}, log u(t,,x)$ must coincide. As a consequence of this fact, we prove that, when $sigma$ is linear, there is a.s. only one such limit point and hence the entire path decays almost surely at an exponential rate.
我们考虑一个随机热方程，$partial_t u = partial^2_x u + sigma(u)dot{W}$在$(0,,infty)times[-1,,1]$上具有周期边界条件和不退化的正初始数据，其中$sigma:mathbb{R} tomathbb{R}$是一个非随机Lipschitz连续函数，$dot{W}$表示时空白噪声。如果另外$sigma(0)=0$，则已知解是严格正的;参见穆勒'91。在这种情况下，我们证明了当时间趋于无穷时，解的对数振荡呈次线性衰减。除其他事项外，可以得出，在概率为1的情况下，$t^{-1}, sup_{xin[-1,1]}, log u(t,,x)$和$t^{-1}, inf_{xin[-1,1]}, log u(t,,x)$的所有极限点必须重合。作为这一事实的结果，我们证明，当$sigma$是线性的，只有一个这样的极限点，因此整个路径几乎肯定以指数速率衰减。
{"title":"Dissipation in parabolic SPDEs II: Oscillation and decay of the solution","authors":"D. Khoshnevisan, Kunwoo Kim, C. Mueller","doi":"10.1214/22-aihp1289","DOIUrl":"https://doi.org/10.1214/22-aihp1289","url":null,"abstract":"We consider a stochastic heat equation of the type, $partial_t u = partial^2_x u + sigma(u)dot{W}$ on $(0,,infty)times[-1,,1]$ with periodic boundary conditions and on-degenerate positive initial data, where $sigma:mathbb{R} tomathbb{R}$ is a non-random Lipschitz continuous function and $dot{W}$ denotes space-time white noise. If additionally $sigma(0)=0$ then the solution is known to be strictly positive; see Mueller '91. In that case, we prove that the oscillation of the logarithm of the solution decays sublinearly as time tends to infinity. Among other things, it follows that, with probability one, all limit points of $t^{-1}, sup_{xin[-1,1]}, log u(t,,x)$ and $t^{-1}, inf_{xin[-1,1]}, log u(t,,x)$ must coincide. As a consequence of this fact, we prove that, when $sigma$ is linear, there is a.s. only one such limit point and hence the entire path decays almost surely at an exponential rate.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"108 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80813508","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}
G. Armstrong, K. Bogdan, T. Grzywny, Lukasz Le.zaj, Longmin Wang
We prove universality of the Yaglom limit of Lipschitz cones among all unimodal L'{e}vy processes sufficiently close to the isotropic $alpha$-stable L'{e}vy process.
{"title":"Yaglom limit for unimodal Lévy processes","authors":"G. Armstrong, K. Bogdan, T. Grzywny, Lukasz Le.zaj, Longmin Wang","doi":"10.1214/22-aihp1301","DOIUrl":"https://doi.org/10.1214/22-aihp1301","url":null,"abstract":"We prove universality of the Yaglom limit of Lipschitz cones among all unimodal L'{e}vy processes sufficiently close to the isotropic $alpha$-stable L'{e}vy process.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"128 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81335998","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 compare the solutions of one-scale Dyson-Schwinger equations in the Minimal subtraction (MS) scheme to the solutions in kinematic (MOM) renormalization schemes. We establish that the MS-solution can be interpreted as a MOM-solution, but with a shifted renormalization point, where the shift itself is a function of the coupling. We derive relations between this shift and various renormalization group functions and counter terms in perturbation theory. As concrete examples, we examine three different one-scale Dyson-Schwinger equations, one based on the D=4 multiedge graph, one for the D=6 multiedge graph and one mathematical toy model. For each of the integral kernels, we examine both the linear and nine different non-linear Dyson-Schwinger equations. For the linear cases, we empirically find exact functional forms of the shift between MOM and MS renormalization points. For the non-linear DSEs, the results for the shift suggest a factorially divergent power series. We determine the leading asymptotic growth parameters and find them in agreement with the ones of the anomalous dimension. Finally, we present a tentative exact non-perturbative solution to one of the non-linear DSEs of the toy model.
{"title":"Dyson–Schwinger equations in minimal subtraction","authors":"Paul-Hermann Balduf","doi":"10.4171/aihpd/169","DOIUrl":"https://doi.org/10.4171/aihpd/169","url":null,"abstract":"We compare the solutions of one-scale Dyson-Schwinger equations in the Minimal subtraction (MS) scheme to the solutions in kinematic (MOM) renormalization schemes. We establish that the MS-solution can be interpreted as a MOM-solution, but with a shifted renormalization point, where the shift itself is a function of the coupling. We derive relations between this shift and various renormalization group functions and counter terms in perturbation theory. As concrete examples, we examine three different one-scale Dyson-Schwinger equations, one based on the D=4 multiedge graph, one for the D=6 multiedge graph and one mathematical toy model. For each of the integral kernels, we examine both the linear and nine different non-linear Dyson-Schwinger equations. For the linear cases, we empirically find exact functional forms of the shift between MOM and MS renormalization points. For the non-linear DSEs, the results for the shift suggest a factorially divergent power series. We determine the leading asymptotic growth parameters and find them in agreement with the ones of the anomalous dimension. Finally, we present a tentative exact non-perturbative solution to one of the non-linear DSEs of the toy model.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"273 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77014636","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}
In this note, we show that the Lyapunov exponents of mixed products of random truncated Haar unitary and complex Ginibre matrices are asymptotically given by equally spaced `picket-fence' statistics. We discuss how these statistics should originate from the connection between random matrix products and multiplicative Brownian motion on $operatorname{GL}_n(mathbb{C})$, analogous to the connection between discrete random walks and ordinary Brownian motion. Our methods are based on contour integral formulas for products of classical matrix ensembles from integrable probability.
{"title":"Lyapunov exponents for truncated unitary and Ginibre matrices","authors":"Andrew Ahn, Roger Van Peski","doi":"10.1214/22-aihp1268","DOIUrl":"https://doi.org/10.1214/22-aihp1268","url":null,"abstract":"In this note, we show that the Lyapunov exponents of mixed products of random truncated Haar unitary and complex Ginibre matrices are asymptotically given by equally spaced `picket-fence' statistics. We discuss how these statistics should originate from the connection between random matrix products and multiplicative Brownian motion on $operatorname{GL}_n(mathbb{C})$, analogous to the connection between discrete random walks and ordinary Brownian motion. Our methods are based on contour integral formulas for products of classical matrix ensembles from integrable probability.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"20 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89657234","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}
A multitype continuous-state branching process (MCSBP) ${rm Z}=({rm Z}_{t})_{tgeq 0}$, is a Markov process with values in $[0,infty)^{d}$ that satisfies the branching property. Its distribution is characterised by its branching mechanism, that is the data of $d$ Laplace exponents of $mathbb{R}^d$-valued spectrally positive L'evy processes, each one having $d-1$ increasing components. We give an expression of the probability for a MCSBP to tend to 0 at infinity in term of its branching mechanism. Then we prove that this extinction holds at a finite time if and only if some condition bearing on the branching mechanism holds. This condition extends Grey's condition that is well known for $d=1$. Our arguments bear on elements of fluctuation theory for spectrally positive additive L'evy fields recently obtained in cite{cma1} and an extension of the Lamperti representation in higher dimension proved in cite{cpgub}.
{"title":"Extinction times of multitype continuous-state branching processes","authors":"L. Chaumont, M. Marolleau","doi":"10.1214/22-aihp1279","DOIUrl":"https://doi.org/10.1214/22-aihp1279","url":null,"abstract":"A multitype continuous-state branching process (MCSBP) ${rm Z}=({rm Z}_{t})_{tgeq 0}$, is a Markov process with values in $[0,infty)^{d}$ that satisfies the branching property. Its distribution is characterised by its branching mechanism, that is the data of $d$ Laplace exponents of $mathbb{R}^d$-valued spectrally positive L'evy processes, each one having $d-1$ increasing components. We give an expression of the probability for a MCSBP to tend to 0 at infinity in term of its branching mechanism. Then we prove that this extinction holds at a finite time if and only if some condition bearing on the branching mechanism holds. This condition extends Grey's condition that is well known for $d=1$. Our arguments bear on elements of fluctuation theory for spectrally positive additive L'evy fields recently obtained in cite{cma1} and an extension of the Lamperti representation in higher dimension proved in cite{cpgub}.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"8 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76016620","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 that the distribution of a Gibbs process with non-negative pair potential is uniquely determined as soon as an associated Poisson-driven random connection model (RCM) does not percolate. Our proof combines disagreement coupling in continuum with a coupling of a Gibbs process and a RCM. The improvement over previous uniqueness results is illustrated both in theory and simulations.
{"title":"On the uniqueness of Gibbs distributions with a non-negative and subcritical pair potential","authors":"Steffen Betsch, G. Last","doi":"10.1214/22-AIHP1265","DOIUrl":"https://doi.org/10.1214/22-AIHP1265","url":null,"abstract":"We prove that the distribution of a Gibbs process with non-negative pair potential is uniquely determined as soon as an associated Poisson-driven random connection model (RCM) does not percolate. Our proof combines disagreement coupling in continuum with a coupling of a Gibbs process and a RCM. The improvement over previous uniqueness results is illustrated both in theory and simulations.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"1 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89621776","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}
P. Eichelsbacher, Benedikt Rednoss, Christoph Thale, Guangqu Zheng
. In this paper, a simplified second-order Gaussian Poincaré inequality for normal approximation of functionals over infinitely many Rademacher random variables is derived. It is based on a new bound for the Kolmogorov distance between a general Rademacher functional and a Gaussian random variable, which is established by means of the discrete Malliavin-Stein method and is of independent interest. As an application, the number of vertices with prescribed degree and the subgraph counting statistic in the Erdős-Rényi random graph are discussed. The number of vertices of fixed degree is also studied for percolation on the Hamming hypercube. Moreover, the number of isolated faces in the Linial-Meshulam-Wallach random κ -complex and infinite weighted 2-runs are treated.
{"title":"A simplified second-order Gaussian Poincaré inequality in discrete setting with applications","authors":"P. Eichelsbacher, Benedikt Rednoss, Christoph Thale, Guangqu Zheng","doi":"10.1214/22-AIHP1247","DOIUrl":"https://doi.org/10.1214/22-AIHP1247","url":null,"abstract":". In this paper, a simplified second-order Gaussian Poincaré inequality for normal approximation of functionals over infinitely many Rademacher random variables is derived. It is based on a new bound for the Kolmogorov distance between a general Rademacher functional and a Gaussian random variable, which is established by means of the discrete Malliavin-Stein method and is of independent interest. As an application, the number of vertices with prescribed degree and the subgraph counting statistic in the Erdős-Rényi random graph are discussed. The number of vertices of fixed degree is also studied for percolation on the Hamming hypercube. Moreover, the number of isolated faces in the Linial-Meshulam-Wallach random κ -complex and infinite weighted 2-runs are treated.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"5 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90547740","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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