{"title":"8 + 1维以上BCC格上最近邻定向渗流的平均场行为","authors":"Lung-Chi Chen, Satoshi Handa, Yoshinori Kamijima","doi":"10.1007/s11040-022-09441-6","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we consider nearest-neighbor oriented percolation with independent Bernoulli bond-occupation probability on the <i>d</i>-dimensional body-centered cubic (BCC) lattice <span>\\({\\mathbb {L}^d}\\)</span> and the set of non-negative integers <span>\\({{\\mathbb {Z}}_+}\\)</span>. Thanks to the orderly structure of the BCC lattice, we prove that the infrared bound holds on <span>\\({\\mathbb {L}^d} \\times {{\\mathbb {Z}}_+}\\)</span> in all dimensions <span>\\(d\\ge 9\\)</span>. As opposed to ordinary percolation, we have to deal with complex numbers due to asymmetry induced by time-orientation, which makes it hard to bound the bootstrap functions in the lace-expansion analysis. By investigating the Fourier–Laplace transform of the random-walk Green function and the two-point function, we derive the key properties to obtain the upper bounds and resolve a problematic issue in Nguyen and Yang’s bound. The issue is caused by the fact that the Fourier transform of the random-walk transition probability can take the value <span>\\(-1\\)</span>.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"26 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Mean-field behavior of Nearest-Neighbor Oriented Percolation on the BCC Lattice Above 8 + 1 Dimensions\",\"authors\":\"Lung-Chi Chen, Satoshi Handa, Yoshinori Kamijima\",\"doi\":\"10.1007/s11040-022-09441-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we consider nearest-neighbor oriented percolation with independent Bernoulli bond-occupation probability on the <i>d</i>-dimensional body-centered cubic (BCC) lattice <span>\\\\({\\\\mathbb {L}^d}\\\\)</span> and the set of non-negative integers <span>\\\\({{\\\\mathbb {Z}}_+}\\\\)</span>. Thanks to the orderly structure of the BCC lattice, we prove that the infrared bound holds on <span>\\\\({\\\\mathbb {L}^d} \\\\times {{\\\\mathbb {Z}}_+}\\\\)</span> in all dimensions <span>\\\\(d\\\\ge 9\\\\)</span>. As opposed to ordinary percolation, we have to deal with complex numbers due to asymmetry induced by time-orientation, which makes it hard to bound the bootstrap functions in the lace-expansion analysis. By investigating the Fourier–Laplace transform of the random-walk Green function and the two-point function, we derive the key properties to obtain the upper bounds and resolve a problematic issue in Nguyen and Yang’s bound. The issue is caused by the fact that the Fourier transform of the random-walk transition probability can take the value <span>\\\\(-1\\\\)</span>.</p></div>\",\"PeriodicalId\":694,\"journal\":{\"name\":\"Mathematical Physics, Analysis and Geometry\",\"volume\":\"26 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-02-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Physics, Analysis and Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s11040-022-09441-6\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Physics, Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s11040-022-09441-6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Mean-field behavior of Nearest-Neighbor Oriented Percolation on the BCC Lattice Above 8 + 1 Dimensions
In this paper, we consider nearest-neighbor oriented percolation with independent Bernoulli bond-occupation probability on the d-dimensional body-centered cubic (BCC) lattice \({\mathbb {L}^d}\) and the set of non-negative integers \({{\mathbb {Z}}_+}\). Thanks to the orderly structure of the BCC lattice, we prove that the infrared bound holds on \({\mathbb {L}^d} \times {{\mathbb {Z}}_+}\) in all dimensions \(d\ge 9\). As opposed to ordinary percolation, we have to deal with complex numbers due to asymmetry induced by time-orientation, which makes it hard to bound the bootstrap functions in the lace-expansion analysis. By investigating the Fourier–Laplace transform of the random-walk Green function and the two-point function, we derive the key properties to obtain the upper bounds and resolve a problematic issue in Nguyen and Yang’s bound. The issue is caused by the fact that the Fourier transform of the random-walk transition probability can take the value \(-1\).
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