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Free boundary dimers: random walk representation and scaling limit. 自由边界二聚体：随机行走表示和缩放限制。
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2023-01-01 Epub Date: 2023-05-16 DOI: 10.1007/s00440-023-01203-x
Nathanaël Berestycki, Marcin Lis, Wei Qian
{"title":"Free boundary dimers: random walk representation and scaling limit.","authors":"Nathanaël Berestycki,&nbsp;Marcin Lis,&nbsp;Wei Qian","doi":"10.1007/s00440-023-01203-x","DOIUrl":"10.1007/s00440-023-01203-x","url":null,"abstract":"<p><p>We study the dimer model on subgraphs of the square lattice in which vertices on a prescribed part of the boundary (the free boundary) are possibly unmatched. Each such unmatched vertex is called a monomer and contributes a fixed multiplicative weight <math><mrow><mi>z</mi><mo>></mo><mn>0</mn></mrow></math> to the total weight of the configuration. A bijection described by Giuliani et al. (J Stat Phys 163(2):211-238, 2016) relates this model to a standard dimer model but on a non-bipartite graph. The Kasteleyn matrix of this dimer model describes a walk with transition weights that are negative along the free boundary. Yet under certain assumptions, which are in particular satisfied in the infinite volume limit in the upper half-plane, we prove an effective, true random walk representation for the inverse Kasteleyn matrix. In this case we further show that, independently of the value of <math><mrow><mi>z</mi><mo>></mo><mn>0</mn></mrow></math>, the scaling limit of the centered height function is the Gaussian free field with Neumann (or free) boundary conditions. It is the first example of a discrete model where such boundary conditions arise in the continuum scaling limit.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"186 3-4","pages":"735-812"},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10271954/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9654894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Global solutions of aggregation equations and other flows with random diffusion. 聚合方程及其他随机扩散流动的全局解。
IF 1.5 1区 数学
Probability Theory and Related Fields Pub Date : 2023-01-01 Epub Date: 2022-10-31 DOI: 10.1007/s00440-022-01171-8
Matthew Rosenzweig, Gigliola Staffilani
{"title":"Global solutions of aggregation equations and other flows with random diffusion.","authors":"Matthew Rosenzweig, Gigliola Staffilani","doi":"10.1007/s00440-022-01171-8","DOIUrl":"10.1007/s00440-022-01171-8","url":null,"abstract":"<p><p>Aggregation equations, such as the parabolic-elliptic Patlak-Keller-Segel model, are known to have an optimal threshold for global existence versus finite-time blow-up. In particular, if the diffusion is absent, then all smooth solutions with finite second moment can exist only locally in time. Nevertheless, one can ask whether global existence can be restored by adding a suitable noise to the equation, so that the dynamics are now stochastic. Inspired by the work of Buckmaster et al. (Int Math Res Not IMRN 23:9370-9385, 2020) showing that, with high probability, the inviscid SQG equation with random diffusion has global classical solutions, we investigate whether suitable random diffusion can restore global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as those arising in aggregation models. For this class, we show global existence of solutions in Gevrey-type Fourier-Lebesgue spaces with quantifiable high probability.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"185 3-4","pages":"1219-1262"},"PeriodicalIF":1.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10032336/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9546081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Periodic Lorentz gas with small scatterers. 具有小散射体的周期洛伦兹气体。
IF 1.5 1区 数学
Probability Theory and Related Fields Pub Date : 2023-01-01 Epub Date: 2023-03-15 DOI: 10.1007/s00440-023-01197-6
Péter Bálint, Henk Bruin, Dalia Terhesiu
{"title":"Periodic Lorentz gas with small scatterers.","authors":"Péter Bálint, Henk Bruin, Dalia Terhesiu","doi":"10.1007/s00440-023-01197-6","DOIUrl":"10.1007/s00440-023-01197-6","url":null,"abstract":"<p><p>We prove limit laws for infinite horizon planar periodic Lorentz gases when, as time <i>n</i> tends to infinity, the scatterer size <math><mi>ρ</mi></math> may also tend to zero simultaneously at a sufficiently slow pace. In particular we obtain a non-standard Central Limit Theorem as well as a Local Limit Theorem for the displacement function. To the best of our knowledge, these are the first results on an intermediate case between the two well-studied regimes with superdiffusive <math><msqrt><mrow><mi>n</mi><mo>log</mo><mi>n</mi></mrow></msqrt></math> scaling (i) for fixed infinite horizon configurations-letting first <math><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></math> and then <math><mrow><mi>ρ</mi><mo>→</mo><mn>0</mn></mrow></math>-studied e.g. by Szász and Varjú (J Stat Phys 129(1):59-80, 2007) and (ii) Boltzmann-Grad type situations-letting first <math><mrow><mi>ρ</mi><mo>→</mo><mn>0</mn></mrow></math> and then <math><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></math>-studied by Marklof and Tóth (Commun Math Phys 347(3):933-981, 2016) .</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"186 1-2","pages":"159-219"},"PeriodicalIF":1.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10169905/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10296939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Biased 2×2 periodic Aztec diamond and an elliptic curve. 偏置的2×2周期阿兹特克钻石和椭圆曲线。
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2023-01-01 Epub Date: 2023-02-14 DOI: 10.1007/s00440-023-01195-8
Alexei Borodin, Maurice Duits
{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Biased <ns0:math><ns0:mrow><ns0:mn>2</ns0:mn><ns0:mo>×</ns0:mo><ns0:mn>2</ns0:mn></ns0:mrow></ns0:math> periodic Aztec diamond and an elliptic curve.","authors":"Alexei Borodin,&nbsp;Maurice Duits","doi":"10.1007/s00440-023-01195-8","DOIUrl":"10.1007/s00440-023-01195-8","url":null,"abstract":"<p><p>We study random domino tilings of the Aztec diamond with a biased <math><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow></math> periodic weight function and associate a linear flow on an elliptic curve to this model. Our main result is a double integral formula for the correlation kernel, in which the integrand is expressed in terms of this flow. For special choices of parameters the flow is periodic, and this allows us to perform a saddle point analysis for the correlation kernel. In these cases we compute the local correlations in the smooth disordered (or gaseous) region. The special example in which the flow has period six is worked out in more detail, and we show that in that case the boundary of the rough disordered region is an algebraic curve of degree eight.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"187 1-2","pages":"259-315"},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10465688/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10129125","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An asymptotic formula for the variance of the number of zeroes of a stationary Gaussian process. 平稳高斯过程零个数方差的一个渐近公式。
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2023-01-01 Epub Date: 2023-09-23 DOI: 10.1007/s00440-023-01218-4
Eran Assaf, Jeremiah Buckley, Naomi Feldheim
{"title":"An asymptotic formula for the variance of the number of zeroes of a stationary Gaussian process.","authors":"Eran Assaf, Jeremiah Buckley, Naomi Feldheim","doi":"10.1007/s00440-023-01218-4","DOIUrl":"https://doi.org/10.1007/s00440-023-01218-4","url":null,"abstract":"<p><p>We study the variance of the number of zeroes of a stationary Gaussian process on a long interval. We give a simple asymptotic description under mild mixing conditions. This allows us to characterise minimal and maximal growth. We show that a small (symmetrised) atom in the spectral measure at a special frequency does not affect the asymptotic growth of the variance, while an atom at any other frequency results in maximal growth.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"187 3-4","pages":"999-1036"},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10628032/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71522465","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Matrix Whittaker processes. Matrix Whittaker过程。
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2023-01-01 Epub Date: 2023-05-14 DOI: 10.1007/s00440-023-01210-y
Jonas Arista, Elia Bisi, Neil O'Connell
{"title":"Matrix Whittaker processes.","authors":"Jonas Arista,&nbsp;Elia Bisi,&nbsp;Neil O'Connell","doi":"10.1007/s00440-023-01210-y","DOIUrl":"10.1007/s00440-023-01210-y","url":null,"abstract":"<p><p>We study a discrete-time Markov process on triangular arrays of matrices of size <math><mrow><mi>d</mi><mo>≥</mo><mn>1</mn></mrow></math>, driven by inverse Wishart random matrices. The components of the right edge evolve as multiplicative random walks on positive definite matrices with one-sided interactions and can be viewed as a <i>d</i>-dimensional generalisation of log-gamma polymer partition functions. We establish intertwining relations to prove that, for suitable initial configurations of the triangular process, the bottom edge has an autonomous Markovian evolution with an explicit transition kernel. We then show that, for a special singular initial configuration, the fixed-time law of the bottom edge is a matrix Whittaker measure, which we define. To achieve this, we perform a Laplace approximation that requires solving a constrained minimisation problem for certain energy functions of matrix arguments on directed graphs.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"187 1-2","pages":"203-257"},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10465476/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10129127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Correction to: Exact value of the resistance exponent for four dimensional random walk trace 修正为:四维随机行走轨迹的电阻指数的精确值
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2022-11-05 DOI: 10.1007/s00440-022-01160-x
D. Croydon, D. Shiraishi
{"title":"Correction to: Exact value of the resistance exponent for four dimensional random walk trace","authors":"D. Croydon, D. Shiraishi","doi":"10.1007/s00440-022-01160-x","DOIUrl":"https://doi.org/10.1007/s00440-022-01160-x","url":null,"abstract":"","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"185 1","pages":"699-704"},"PeriodicalIF":2.0,"publicationDate":"2022-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47729417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Existence of strong solutions for Itô’s stochastic equations via approximations: revisited Itô随机方程的近似强解的存在性
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2022-09-01 DOI: 10.1007/s40072-022-00273-7
I. Gyöngy, N. V. Krylov
{"title":"Existence of strong solutions for Itô’s stochastic equations via approximations: revisited","authors":"I. Gyöngy, N. V. Krylov","doi":"10.1007/s40072-022-00273-7","DOIUrl":"https://doi.org/10.1007/s40072-022-00273-7","url":null,"abstract":"","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"10 1","pages":"693 - 719"},"PeriodicalIF":2.0,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52759041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Sandwiching dense random regular graphs between binomial random graphs 将密集随机正则图夹在二项随机图之间
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2022-08-06 DOI: 10.1007/s00440-022-01157-6
Pu Gao, M. Isaev, B. McKay
{"title":"Sandwiching dense random regular graphs between binomial random graphs","authors":"Pu Gao, M. Isaev, B. McKay","doi":"10.1007/s00440-022-01157-6","DOIUrl":"https://doi.org/10.1007/s00440-022-01157-6","url":null,"abstract":"","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"184 1","pages":"115 - 158"},"PeriodicalIF":2.0,"publicationDate":"2022-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42911006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 16
Quasi-stationary distribution for Hamiltonian dynamics with singular potentials 奇异势哈密顿动力学的拟平稳分布
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2022-07-13 DOI: 10.1007/s00440-022-01154-9
A. Guillin, Boris Nectoux, Liming Wu
{"title":"Quasi-stationary distribution for Hamiltonian dynamics with singular potentials","authors":"A. Guillin, Boris Nectoux, Liming Wu","doi":"10.1007/s00440-022-01154-9","DOIUrl":"https://doi.org/10.1007/s00440-022-01154-9","url":null,"abstract":"","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"185 1","pages":"921 - 959"},"PeriodicalIF":2.0,"publicationDate":"2022-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43547191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
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