Intersections of random walks and Wiener sausages in four dimensions |
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Authors: | Sergio Albeverio Xian Yin Zhou |
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Affiliation: | (1) Institute of Mathematics, Ruhr-University Bochum, Postfach 102148, D-44780 Bochum 1, Germany;(2) BiBoS, SFB 237 Bochum-Essen-Düsseldorf;(3) CERFIM, Locarno, Switzerland;(4) Department of Mathematics, Beijing Normal University, 100875 Beijing, P.R. China;(5) Institute of Mathematics, Ruhr-University Bochum, Postfach 102148, D-44780 Bochum 1, Germany |
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Abstract: | Let {S1(n)}n0and {S2(n)}n0be independent simple random walks in Z4 starting at the origin, and let % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfc6aqnaaBaaaleaacaqGPbaabeaatCvAUfKttLearyqr1ngB% Prgaiuaakiab-HcaOiaadggacaGGSaGaamOyaiab-LcaPiabg2da9i% ab-Tha7Hqbciab+Hha4jabgIGiolab+PfaAnaaCaaaleqabaGaaGin% aaaakiaacQdaieGacaqFtbWaaSbaaSqaaiaabMgaaeqaaOGae8hkaG% Iaa0xBaiab-LcaPiabg2da9iab+Hha4baa!5761![Pi _{rm{i}} (a,b) = { x in Z^4 :S_{rm{i}} (m) = x]for the some % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciaa-1gacqGHiiIZtCvAUfKttLearyqr1ngBPrgaiuaacqGF% OaakcaWGHbGaaiilaiaadkgacqGFPaqkcqGF9bqFaaa!4936![m in (a,b)} ]. Let two integervalued sequences {an}n0and {bn}n0be given, such that the limit limn anexists and limn bn=+. In this paper, it is shown that the probability of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfc6aqnaaBaaaleaacaaIXaaabeaatCvAUfKttLearyqr1ngB% Prgaiuaakiab-HcaOiab-bdaWiab-XcaSiabg6HiLkab-LcaPiabgM% Iihlabfc6aqnaaBaaaleaacaaIYaaabeaakiab-HcaOiaadggadaWg% aaWcbaGaamOBaiaacYcaaeqaaOGaamyyamaaBaaaleaacaWGUbaabe% aakiabgUcaRiaadkgadaWgaaWcbaGaamOBaaqabaGccqWFPaqkcqGH% GjsUieaacaGFydaaaa!5904![Pi _1 (0,infty ) cap Pi _2 (a_{n,} a_n + b_n ) ne O ] is asymptotic to % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaalaaabaGaaGymaaqaaiaaikdaaaGaciiBaiaac+gacaGGNbWe% xLMBb50ujbqeguuDJXwAKbacfaGae8hkaGIae8xmaeJae83kaSIaam% OyamaaBaaaleaacaWGUbaabeaakiaac+cacaWGHbWaaSbaaSqaaiaa% d6gaaeqaaOGae8xkaKIae83la8IaciiBaiaac+gacaGGNbGaamOyam% aaBaaaleaacaWGUbaabeaaaaa!5364![frac{1}{2}log (1 + b_n /a_n )/log b_n ] if it tends to zero as n, and the probability of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfc6aqnaaBaaaleaacaaIXaaabeaatCvAUfKttLearyqr1ngB% Prgaiuaakiab-HcaOiab-bdaWiab-XcaSiabg6HiLkab-LcaPiabgM% Iihlabfc6aqnaaBaaaleaacaaIYaaabeaakiab-HcaOiaadggadaWg% aaWcbaGaamOBaaqabaGccaGGSaGaamyyamaaBaaaleaacaWGUbaabe% aakiabgUcaRiaadkgadaWgaaWcbaGaamOBaaqabaGccqWFPaqkcqWF% 9aqpieaacaGFydaaaa!583C![Pi _1 (0,infty ) cap Pi _2 (a_n ,a_n + b_n ) = O ]is asymptotic to % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% abaeqabaaabaGaam4yaiaacUfaciGGSbGaai4BaiaacEgatCvAUfKt% tLearyqr1ngBPrgaiuaacqWFOaakcaWGHbWaaSbaaSqaaiaad6gaae% qaaOGaey4kaSIaamOyamaaBaaaleaacaWGUbaabeaakiab-LcaPiab% -9caViab-XgaSjab-9gaVjab-DgaNjab-HcaOiaadggadaWgaaWcba% GaamOBaaqabaGccqGHRaWkcaaIYaGae8xkaKIae8xxa01aaWbaaSqa% beaacqWFTaqlcqWFXaqmcqWFVaWlcqWFYaGmaaaaaaa!5BAC![begin{array}{l} Pi _1 (0,infty ) cap Pi _2 (a_n ,a_n + b_n ) = O c[log (a_n + b_n )/log(a_n + 2)]^{ - 1/2} end{array}], for some constant c, if it tends to a finite constant (1) as n. These results extend some results obtained by G. F. Lawler about the intersection properties of simple random walks in Z4. By using similar arguments, we also get corresponding results for the intersections of Wiener sausages in four dimensions. In particular, a conjecture suggested by M. Aizenman, which describes nonintersection of independent Wiener sausages in R4, is proven.Partly supported by AvH Foundation. |
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Keywords: | 60G17 60J15 60J65 |
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