Random walk on the zero-range process
Autor(a) principal: | |
---|---|
Data de Publicação: | 2022 |
Tipo de documento: | Tese |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UFMG |
Texto Completo: | http://hdl.handle.net/1843/50995 https://orcid.org/0000-0003-4875-3805 |
Resumo: | The main theme of this thesis is the study of random walks in random environments. More specifically, we will study a random walk that moves in a medium that is inhomogeneous in $\mathbb{Z}$ and evolves with time, called dynamic random environment. In this case, the environment will be an interacting particle system in $\mathbb{Z}$ known as the zero-range process. This name comes from the fact that, in this system, particles interact with each other only when they share the same site. We will denote this process by $\eta=(\eta_{t})_{t\geq 0}$. We will assume that this process is in equilibrium. We say that site $x \in \mathbb{Z}$ is occupied by particles at time $t$ if $\eta_{t}(x)>0$ and othewise, we say that $x$ is vacant. Let $\alpha_{\bullet}, \beta_{\bullet}, \alpha_{\circ}$ and $\beta_{\circ}$ be positive real numbers. On $\eta$ we define a continuous-time, nearest-neighbor random walk $X=(X_t)_{t \geq 0}$, that starts at the origin and evolves as follows: when $X$ is on a site occupied by particles, it jumps to the right with a rate $\alpha_{\bullet}$ and to the left with a rate $\beta_{\bullet}$; when $X$ is on a vacant site, these rates are respectively $\alpha_{\circ}$ and $\beta_{\circ}$. Here, the random walk does not interfere with the evolution of the environment. We will study the asymptotic behavior of this random walk in time in the case where $X$ has a positive local drift to the right. More precisely, assuming that the differences $\alpha_{\bullet}-\beta_{\bullet}$ and $\alpha_{\circ}-\beta_{\circ}$ are sufficiently large, we will obtain a law of large numbers for $X $. That is, we will show that $X_t/t$ converges almost surely to a value $v$, which depends of the density of particles of the environment. Random walks in dynamic random environments given by interacting particle system have attracted a lot of interest in the last two decades. Several techniques have been developed and used to obtain refined results about these processes such as laws of large numbers, central limit theorems, invariance principles and large deviation estimates. Some of these techniques are regeneration, analytical techniques and multiscale renormalization. The applicability of each of the techniques and the range of results depend on the mixing properties of the environment. In this context, conservative particle systems such as the exclusion process, independent random walks and the zero-range process have drawn much attention due to the lack of good mixing properties. Thus, we hope that the results in this thesis contribute on the mathematical understanding of these processes. |
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Marcelo Richard Hiláriohttp://lattes.cnpq.br/2075091409733505Renato Soares dos SantosAugusto TeixeiraBernardo Nunes Borges de LimaLuca AvenaTertuliano FrancoRangel Baldassohttp://lattes.cnpq.br/2271787950677680Weberson da Silva Arcanjo2023-03-17T15:54:32Z2023-03-17T15:54:32Z2022-03-11http://hdl.handle.net/1843/50995https://orcid.org/0000-0003-4875-3805The main theme of this thesis is the study of random walks in random environments. More specifically, we will study a random walk that moves in a medium that is inhomogeneous in $\mathbb{Z}$ and evolves with time, called dynamic random environment. In this case, the environment will be an interacting particle system in $\mathbb{Z}$ known as the zero-range process. This name comes from the fact that, in this system, particles interact with each other only when they share the same site. We will denote this process by $\eta=(\eta_{t})_{t\geq 0}$. We will assume that this process is in equilibrium. We say that site $x \in \mathbb{Z}$ is occupied by particles at time $t$ if $\eta_{t}(x)>0$ and othewise, we say that $x$ is vacant. Let $\alpha_{\bullet}, \beta_{\bullet}, \alpha_{\circ}$ and $\beta_{\circ}$ be positive real numbers. On $\eta$ we define a continuous-time, nearest-neighbor random walk $X=(X_t)_{t \geq 0}$, that starts at the origin and evolves as follows: when $X$ is on a site occupied by particles, it jumps to the right with a rate $\alpha_{\bullet}$ and to the left with a rate $\beta_{\bullet}$; when $X$ is on a vacant site, these rates are respectively $\alpha_{\circ}$ and $\beta_{\circ}$. Here, the random walk does not interfere with the evolution of the environment. We will study the asymptotic behavior of this random walk in time in the case where $X$ has a positive local drift to the right. More precisely, assuming that the differences $\alpha_{\bullet}-\beta_{\bullet}$ and $\alpha_{\circ}-\beta_{\circ}$ are sufficiently large, we will obtain a law of large numbers for $X $. That is, we will show that $X_t/t$ converges almost surely to a value $v$, which depends of the density of particles of the environment. Random walks in dynamic random environments given by interacting particle system have attracted a lot of interest in the last two decades. Several techniques have been developed and used to obtain refined results about these processes such as laws of large numbers, central limit theorems, invariance principles and large deviation estimates. Some of these techniques are regeneration, analytical techniques and multiscale renormalization. The applicability of each of the techniques and the range of results depend on the mixing properties of the environment. In this context, conservative particle systems such as the exclusion process, independent random walks and the zero-range process have drawn much attention due to the lack of good mixing properties. Thus, we hope that the results in this thesis contribute on the mathematical understanding of these processes.O tema principal desta tese é o estudo de passeios aleatórios em ambientes aleatórios. Mais especificamente, estudaremos um passeio aleatório que se move em um meio que é não-homogêneo em $\mathbb{Z}$ e se modifica ao longo do tempo, denominado ambiente aleatório dinâmico. Neste caso, o ambiente será um sistema de partículas interagentes em $\mathbb{Z}$ conhecido como processo \textit{zero-range}. O nome vem do fato de que, nesse sistema, as partículas interagem umas com as outras somente quando estão no mesmo sítio. Denotaremos este processo por $\eta=(\eta_{t})_{t \geq 0}$. Vamos assumir que o ambiente esteja em equilíbrio. Diremos que o sítio $x \in \mathbb{Z}$ está ocupado por partículas no tempo $t$ se $\eta_{t}(x)>0$ e, caso contrário, diremos que $x$ está vacante. Sejam $\alpha_{\bullet}, \beta_{\bullet}, \alpha_{\circ}$ e $\beta_{\circ}$ números reais positivos. Sobre $\eta$ definimos um passeio \linebreak aleatório $X=(X_t)_{t \geq 0}$, a tempo contínuo e com saltos de primeiros vizinhos, que começa na origem e evolui da seguinte maneira: quando $X$ se encontra sobre um sítio ocupado por partículas, ele salta para a direita com uma taxa $\alpha_{\bullet}$ e para a esquerda com uma taxa $\beta_{\bullet}$; quando $X$ está sobre um sítio vacante, essas taxas são respectivamente $\alpha_{\circ}$ e $\beta_{\circ}$. Aqui, o passeio aleatório não interfere na evolução do ambiente. Estudaremos o comportamento assintótico deste passeio aleatório ao longo do tempo no caso em que $X$ possui um \textit{drift} local positivo para a direita. Mais precisamente, supondo que as diferenças $\alpha_{\bullet}-\beta_{\bullet}$ e $\alpha_{\circ}-\beta_{\circ}$ sejam suficientemente grandes, obteremos uma lei dos grandes números para $X$. Isto é, mostraremos que $X_t/t$ converge quase certamente para um valor $v$, que depende da densidade de partículas do ambiente. Passeios aleatórios em ambientes aleatórios dinâmicos dados por sistema de partículas interagentes têm atraído bastante interesse nas últimas duas décadas. Várias técnicas foram desenvolvidas e usadas para obter resultados refinados sobre esses processos como leis dos grandes números, teoremas centrais do limite, princípios de invariância e estimativas de grandes desvios. Algumas dessas técnicas são regeneração, técnicas analíticas e renormalização em multi-escala. A aplicabilidade de cada uma das técnicas e o alcance dos resultados dependem das propriedades de mistura (mixing) do ambiente. Neste contexto, sistemas de partículas conservativos como o processo de exclusão, passeios aleatórios independentes e o processo zero-range atraíram muita atenção devido à falta de boas propriedades de mistura. Assim, esperamos que os resultados nesta tese contribuam para a compreensão matemática desses processos.CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorengUniversidade Federal de Minas GeraisPrograma de Pós-Graduação em MatemáticaUFMGBrasilICX - DEPARTAMENTO DE MATEMÁTICAMatemática – TesesPasseio aleatório (Matemática) – TesesLei dos Grandes Números – TesesRandom WalkZero-RangeLaw of Large NumbersRandom walk on the zero-range processinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/openAccessreponame:Repositório Institucional da UFMGinstname:Universidade Federal de Minas Gerais (UFMG)instacron:UFMGORIGINALTeseWeb.pdfTeseWeb.pdfapplication/pdf1383387https://repositorio.ufmg.br/bitstream/1843/50995/3/TeseWeb.pdf2eadb31f055c8d93615ca8453941da92MD53LICENSElicense.txtlicense.txttext/plain; charset=utf-82118https://repositorio.ufmg.br/bitstream/1843/50995/4/license.txtcda590c95a0b51b4d15f60c9642ca272MD541843/509952023-03-17 12:54:33.226oai:repositorio.ufmg.br:1843/50995TElDRU7Dh0EgREUgRElTVFJJQlVJw4fDg08gTsODTy1FWENMVVNJVkEgRE8gUkVQT1NJVMOTUklPIElOU1RJVFVDSU9OQUwgREEgVUZNRwoKQ29tIGEgYXByZXNlbnRhw6fDo28gZGVzdGEgbGljZW7Dp2EsIHZvY8OqIChvIGF1dG9yIChlcykgb3UgbyB0aXR1bGFyIGRvcyBkaXJlaXRvcyBkZSBhdXRvcikgY29uY2VkZSBhbyBSZXBvc2l0w7NyaW8gSW5zdGl0dWNpb25hbCBkYSBVRk1HIChSSS1VRk1HKSBvIGRpcmVpdG8gbsOjbyBleGNsdXNpdm8gZSBpcnJldm9nw6F2ZWwgZGUgcmVwcm9kdXppciBlL291IGRpc3RyaWJ1aXIgYSBzdWEgcHVibGljYcOnw6NvIChpbmNsdWluZG8gbyByZXN1bW8pIHBvciB0b2RvIG8gbXVuZG8gbm8gZm9ybWF0byBpbXByZXNzbyBlIGVsZXRyw7RuaWNvIGUgZW0gcXVhbHF1ZXIgbWVpbywgaW5jbHVpbmRvIG9zIGZvcm1hdG9zIMOhdWRpbyBvdSB2w61kZW8uCgpWb2PDqiBkZWNsYXJhIHF1ZSBjb25oZWNlIGEgcG9sw610aWNhIGRlIGNvcHlyaWdodCBkYSBlZGl0b3JhIGRvIHNldSBkb2N1bWVudG8gZSBxdWUgY29uaGVjZSBlIGFjZWl0YSBhcyBEaXJldHJpemVzIGRvIFJJLVVGTUcuCgpWb2PDqiBjb25jb3JkYSBxdWUgbyBSZXBvc2l0w7NyaW8gSW5zdGl0dWNpb25hbCBkYSBVRk1HIHBvZGUsIHNlbSBhbHRlcmFyIG8gY29udGXDumRvLCB0cmFuc3BvciBhIHN1YSBwdWJsaWNhw6fDo28gcGFyYSBxdWFscXVlciBtZWlvIG91IGZvcm1hdG8gcGFyYSBmaW5zIGRlIHByZXNlcnZhw6fDo28uCgpWb2PDqiB0YW1iw6ltIGNvbmNvcmRhIHF1ZSBvIFJlcG9zaXTDs3JpbyBJbnN0aXR1Y2lvbmFsIGRhIFVGTUcgcG9kZSBtYW50ZXIgbWFpcyBkZSB1bWEgY8OzcGlhIGRlIHN1YSBwdWJsaWNhw6fDo28gcGFyYSBmaW5zIGRlIHNlZ3VyYW7Dp2EsIGJhY2stdXAgZSBwcmVzZXJ2YcOnw6NvLgoKVm9jw6ogZGVjbGFyYSBxdWUgYSBzdWEgcHVibGljYcOnw6NvIMOpIG9yaWdpbmFsIGUgcXVlIHZvY8OqIHRlbSBvIHBvZGVyIGRlIGNvbmNlZGVyIG9zIGRpcmVpdG9zIGNvbnRpZG9zIG5lc3RhIGxpY2Vuw6dhLiBWb2PDqiB0YW1iw6ltIGRlY2xhcmEgcXVlIG8gZGVww7NzaXRvIGRlIHN1YSBwdWJsaWNhw6fDo28gbsOjbywgcXVlIHNlamEgZGUgc2V1IGNvbmhlY2ltZW50bywgaW5mcmluZ2UgZGlyZWl0b3MgYXV0b3JhaXMgZGUgbmluZ3XDqW0uCgpDYXNvIGEgc3VhIHB1YmxpY2HDp8OjbyBjb250ZW5oYSBtYXRlcmlhbCBxdWUgdm9jw6ogbsOjbyBwb3NzdWkgYSB0aXR1bGFyaWRhZGUgZG9zIGRpcmVpdG9zIGF1dG9yYWlzLCB2b2PDqiBkZWNsYXJhIHF1ZSBvYnRldmUgYSBwZXJtaXNzw6NvIGlycmVzdHJpdGEgZG8gZGV0ZW50b3IgZG9zIGRpcmVpdG9zIGF1dG9yYWlzIHBhcmEgY29uY2VkZXIgYW8gUmVwb3NpdMOzcmlvIEluc3RpdHVjaW9uYWwgZGEgVUZNRyBvcyBkaXJlaXRvcyBhcHJlc2VudGFkb3MgbmVzdGEgbGljZW7Dp2EsIGUgcXVlIGVzc2UgbWF0ZXJpYWwgZGUgcHJvcHJpZWRhZGUgZGUgdGVyY2Vpcm9zIGVzdMOhIGNsYXJhbWVudGUgaWRlbnRpZmljYWRvIGUgcmVjb25oZWNpZG8gbm8gdGV4dG8gb3Ugbm8gY29udGXDumRvIGRhIHB1YmxpY2HDp8OjbyBvcmEgZGVwb3NpdGFkYS4KCkNBU08gQSBQVUJMSUNBw4fDg08gT1JBIERFUE9TSVRBREEgVEVOSEEgU0lETyBSRVNVTFRBRE8gREUgVU0gUEFUUk9Dw41OSU8gT1UgQVBPSU8gREUgVU1BIEFHw4pOQ0lBIERFIEZPTUVOVE8gT1UgT1VUUk8gT1JHQU5JU01PLCBWT0PDiiBERUNMQVJBIFFVRSBSRVNQRUlUT1UgVE9ET1MgRSBRVUFJU1FVRVIgRElSRUlUT1MgREUgUkVWSVPDg08gQ09NTyBUQU1Cw4lNIEFTIERFTUFJUyBPQlJJR0HDh8OVRVMgRVhJR0lEQVMgUE9SIENPTlRSQVRPIE9VIEFDT1JETy4KCk8gUmVwb3NpdMOzcmlvIEluc3RpdHVjaW9uYWwgZGEgVUZNRyBzZSBjb21wcm9tZXRlIGEgaWRlbnRpZmljYXIgY2xhcmFtZW50ZSBvIHNldSBub21lKHMpIG91IG8ocykgbm9tZXMocykgZG8ocykgZGV0ZW50b3IoZXMpIGRvcyBkaXJlaXRvcyBhdXRvcmFpcyBkYSBwdWJsaWNhw6fDo28sIGUgbsOjbyBmYXLDoSBxdWFscXVlciBhbHRlcmHDp8OjbywgYWzDqW0gZGFxdWVsYXMgY29uY2VkaWRhcyBwb3IgZXN0YSBsaWNlbsOnYS4KRepositório de PublicaçõesPUBhttps://repositorio.ufmg.br/oaiopendoar:2023-03-17T15:54:33Repositório Institucional da UFMG - Universidade Federal de Minas Gerais (UFMG)false |
dc.title.pt_BR.fl_str_mv |
Random walk on the zero-range process |
title |
Random walk on the zero-range process |
spellingShingle |
Random walk on the zero-range process Weberson da Silva Arcanjo Random Walk Zero-Range Law of Large Numbers Matemática – Teses Passeio aleatório (Matemática) – Teses Lei dos Grandes Números – Teses |
title_short |
Random walk on the zero-range process |
title_full |
Random walk on the zero-range process |
title_fullStr |
Random walk on the zero-range process |
title_full_unstemmed |
Random walk on the zero-range process |
title_sort |
Random walk on the zero-range process |
author |
Weberson da Silva Arcanjo |
author_facet |
Weberson da Silva Arcanjo |
author_role |
author |
dc.contributor.advisor1.fl_str_mv |
Marcelo Richard Hilário |
dc.contributor.advisor1Lattes.fl_str_mv |
http://lattes.cnpq.br/2075091409733505 |
dc.contributor.advisor-co1.fl_str_mv |
Renato Soares dos Santos |
dc.contributor.referee1.fl_str_mv |
Augusto Teixeira |
dc.contributor.referee2.fl_str_mv |
Bernardo Nunes Borges de Lima |
dc.contributor.referee3.fl_str_mv |
Luca Avena |
dc.contributor.referee4.fl_str_mv |
Tertuliano Franco |
dc.contributor.referee5.fl_str_mv |
Rangel Baldasso |
dc.contributor.authorLattes.fl_str_mv |
http://lattes.cnpq.br/2271787950677680 |
dc.contributor.author.fl_str_mv |
Weberson da Silva Arcanjo |
contributor_str_mv |
Marcelo Richard Hilário Renato Soares dos Santos Augusto Teixeira Bernardo Nunes Borges de Lima Luca Avena Tertuliano Franco Rangel Baldasso |
dc.subject.por.fl_str_mv |
Random Walk Zero-Range Law of Large Numbers |
topic |
Random Walk Zero-Range Law of Large Numbers Matemática – Teses Passeio aleatório (Matemática) – Teses Lei dos Grandes Números – Teses |
dc.subject.other.pt_BR.fl_str_mv |
Matemática – Teses Passeio aleatório (Matemática) – Teses Lei dos Grandes Números – Teses |
description |
The main theme of this thesis is the study of random walks in random environments. More specifically, we will study a random walk that moves in a medium that is inhomogeneous in $\mathbb{Z}$ and evolves with time, called dynamic random environment. In this case, the environment will be an interacting particle system in $\mathbb{Z}$ known as the zero-range process. This name comes from the fact that, in this system, particles interact with each other only when they share the same site. We will denote this process by $\eta=(\eta_{t})_{t\geq 0}$. We will assume that this process is in equilibrium. We say that site $x \in \mathbb{Z}$ is occupied by particles at time $t$ if $\eta_{t}(x)>0$ and othewise, we say that $x$ is vacant. Let $\alpha_{\bullet}, \beta_{\bullet}, \alpha_{\circ}$ and $\beta_{\circ}$ be positive real numbers. On $\eta$ we define a continuous-time, nearest-neighbor random walk $X=(X_t)_{t \geq 0}$, that starts at the origin and evolves as follows: when $X$ is on a site occupied by particles, it jumps to the right with a rate $\alpha_{\bullet}$ and to the left with a rate $\beta_{\bullet}$; when $X$ is on a vacant site, these rates are respectively $\alpha_{\circ}$ and $\beta_{\circ}$. Here, the random walk does not interfere with the evolution of the environment. We will study the asymptotic behavior of this random walk in time in the case where $X$ has a positive local drift to the right. More precisely, assuming that the differences $\alpha_{\bullet}-\beta_{\bullet}$ and $\alpha_{\circ}-\beta_{\circ}$ are sufficiently large, we will obtain a law of large numbers for $X $. That is, we will show that $X_t/t$ converges almost surely to a value $v$, which depends of the density of particles of the environment. Random walks in dynamic random environments given by interacting particle system have attracted a lot of interest in the last two decades. Several techniques have been developed and used to obtain refined results about these processes such as laws of large numbers, central limit theorems, invariance principles and large deviation estimates. Some of these techniques are regeneration, analytical techniques and multiscale renormalization. The applicability of each of the techniques and the range of results depend on the mixing properties of the environment. In this context, conservative particle systems such as the exclusion process, independent random walks and the zero-range process have drawn much attention due to the lack of good mixing properties. Thus, we hope that the results in this thesis contribute on the mathematical understanding of these processes. |
publishDate |
2022 |
dc.date.issued.fl_str_mv |
2022-03-11 |
dc.date.accessioned.fl_str_mv |
2023-03-17T15:54:32Z |
dc.date.available.fl_str_mv |
2023-03-17T15:54:32Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
format |
doctoralThesis |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/1843/50995 |
dc.identifier.orcid.pt_BR.fl_str_mv |
https://orcid.org/0000-0003-4875-3805 |
url |
http://hdl.handle.net/1843/50995 https://orcid.org/0000-0003-4875-3805 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Universidade Federal de Minas Gerais |
dc.publisher.program.fl_str_mv |
Programa de Pós-Graduação em Matemática |
dc.publisher.initials.fl_str_mv |
UFMG |
dc.publisher.country.fl_str_mv |
Brasil |
dc.publisher.department.fl_str_mv |
ICX - DEPARTAMENTO DE MATEMÁTICA |
publisher.none.fl_str_mv |
Universidade Federal de Minas Gerais |
dc.source.none.fl_str_mv |
reponame:Repositório Institucional da UFMG instname:Universidade Federal de Minas Gerais (UFMG) instacron:UFMG |
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UFMG |
institution |
UFMG |
reponame_str |
Repositório Institucional da UFMG |
collection |
Repositório Institucional da UFMG |
bitstream.url.fl_str_mv |
https://repositorio.ufmg.br/bitstream/1843/50995/3/TeseWeb.pdf https://repositorio.ufmg.br/bitstream/1843/50995/4/license.txt |
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repository.name.fl_str_mv |
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