| spelling |
Roger William Câmara Silvahttp://lattes.cnpq.br/2131063265034220Sokol NdrecaBernardo Nunes Borges de Limahttp://lattes.cnpq.br/4585474317607896André Victor Ribeiro Amaral2023-03-29T14:27:14Z2023-03-29T14:27:14Z2020-08-10http://hdl.handle.net/1843/51318https://orcid.org/0000-0003-3748-6801O conceito de transição de fase, inerente a diferentes tipos de modelos puramente determinísticos, pode, também, ser verificado em sistemas com componentes estocásticas. Nesse sentido, é importante o estudo de ferramentas que nos permitem provar que determinados sistemas aleatórios apresentam esse tipo de característica. Assim, dado um espaço de probabilidade apropriado, resultados associados à análise de funções Booleanas desempenham papel importante no estudo dessa classe de modelos. Este texto preocupa-se em, ao longo da Seção 2, apresentar e demonstrar tais resultados. Tratando-se de modelos aleatórios independentes e que vêm da Física Estatística, o de Percolação Bernoulli é, talvez, o mais conhecido. Por isso, na Seção 3, nos concentramos em reproduzir alguns resultados “clássicos” desse modelo; utilizando, porém, as ferramentas desenvolvidas na Seção 2 Aqui, é importante ressaltar o ganho que existe em utilizar esse tipo de abordagem. Alguns dos resultados demonstrados poderão ser estendidos, utilizando-se de estratégias similares, para modelos construídos sobre espaços mais gerais – incluindo modelos com dependência, como os que foram discutidos na Seção 4. Por fim, deixo o registro de que os resultados apresentados ao longo desse texto não são originais. O trabalho foi construído a partir de, principalmente, Duminil-Copin (2019) – em adição aos outros artigos e recursos que foram apropriadamente referenciados.The phase transition concept, intrinsic to some purely deterministic models, may also be seen in systems with some stochastic component. Therefore, it is important to study results that allow us to prove that some arbitrary random systems present such a characteristic. In this regard, given an appropriate probability space, results associated with Boolean function analysis play an important role in the study of this class of models. As a consequence of it, this text focuses on, throughout Section 2, introducing and proving such results. With respect to independent random models from Statistical Physics, the Bernoulli Percolation Model may be considered the most popular one. Thus, in Section 3, we focused on replicating some “classical” results concerned with it. In order to achieve this, we used the tools developed in Section 2. At this point, it is important to stress the benefits of adopting such an approach. Some of the proofs may be extended, through similar strategies, to models defined over more general spaces – which also includes models with a structure of dependence, as discussed in Section 4. Finally, I would like to clarify that the results presented throughout this text are not original. This work was developed mainly based on Duminil-Copin (2019) – in addition to other resources and academic articles, which were properly cited.CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorporUniversidade Federal de Minas GeraisPrograma de Pós-Graduação em EstatísticaUFMGBrasilICX - DEPARTAMENTO DE ESTATÍSTICAEstatística - TesesFísica estatística.PercolaçãoTransição de faseAnálise de funções booleanas.PercolaçãoTransição de faseAnálise de funções booleanasTransição de fase em modelos de percolação via funções booleanasPhase transition phenomenon in percolation models using Boolean functionsinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/openAccessreponame:Repositório Institucional da UFMGinstname:Universidade Federal de Minas Gerais (UFMG)instacron:UFMGORIGINALDISSERTAÇÃO.pdfDISSERTAÇÃO.pdfDissertaçãoapplication/pdf693209https://repositorio.ufmg.br/bitstream/1843/51318/5/DISSERTAC%cc%a7A%cc%83O.pdf8acccbdc33198ff68917dd1dc64c7641MD55LICENSElicense.txtlicense.txttext/plain; charset=utf-82118https://repositorio.ufmg.br/bitstream/1843/51318/6/license.txtcda590c95a0b51b4d15f60c9642ca272MD561843/513182023-03-29 11:27:15.015oai:repositorio.ufmg.br:1843/51318TElDRU7Dh0EgREUgRElTVFJJQlVJw4fDg08gTsODTy1FWENMVVNJVkEgRE8gUkVQT1NJVMOTUklPIElOU1RJVFVDSU9OQUwgREEgVUZNRwoKQ29tIGEgYXByZXNlbnRhw6fDo28gZGVzdGEgbGljZW7Dp2EsIHZvY8OqIChvIGF1dG9yIChlcykgb3UgbyB0aXR1bGFyIGRvcyBkaXJlaXRvcyBkZSBhdXRvcikgY29uY2VkZSBhbyBSZXBvc2l0w7NyaW8gSW5zdGl0dWNpb25hbCBkYSBVRk1HIChSSS1VRk1HKSBvIGRpcmVpdG8gbsOjbyBleGNsdXNpdm8gZSBpcnJldm9nw6F2ZWwgZGUgcmVwcm9kdXppciBlL291IGRpc3RyaWJ1aXIgYSBzdWEgcHVibGljYcOnw6NvIChpbmNsdWluZG8gbyByZXN1bW8pIHBvciB0b2RvIG8gbXVuZG8gbm8gZm9ybWF0byBpbXByZXNzbyBlIGVsZXRyw7RuaWNvIGUgZW0gcXVhbHF1ZXIgbWVpbywgaW5jbHVpbmRvIG9zIGZvcm1hdG9zIMOhdWRpbyBvdSB2w61kZW8uCgpWb2PDqiBkZWNsYXJhIHF1ZSBjb25oZWNlIGEgcG9sw610aWNhIGRlIGNvcHlyaWdodCBkYSBlZGl0b3JhIGRvIHNldSBkb2N1bWVudG8gZSBxdWUgY29uaGVjZSBlIGFjZWl0YSBhcyBEaXJldHJpemVzIGRvIFJJLVVGTUcuCgpWb2PDqiBjb25jb3JkYSBxdWUgbyBSZXBvc2l0w7NyaW8gSW5zdGl0dWNpb25hbCBkYSBVRk1HIHBvZGUsIHNlbSBhbHRlcmFyIG8gY29udGXDumRvLCB0cmFuc3BvciBhIHN1YSBwdWJsaWNhw6fDo28gcGFyYSBxdWFscXVlciBtZWlvIG91IGZvcm1hdG8gcGFyYSBmaW5zIGRlIHByZXNlcnZhw6fDo28uCgpWb2PDqiB0YW1iw6ltIGNvbmNvcmRhIHF1ZSBvIFJlcG9zaXTDs3JpbyBJbnN0aXR1Y2lvbmFsIGRhIFVGTUcgcG9kZSBtYW50ZXIgbWFpcyBkZSB1bWEgY8OzcGlhIGRlIHN1YSBwdWJsaWNhw6fDo28gcGFyYSBmaW5zIGRlIHNlZ3VyYW7Dp2EsIGJhY2stdXAgZSBwcmVzZXJ2YcOnw6NvLgoKVm9jw6ogZGVjbGFyYSBxdWUgYSBzdWEgcHVibGljYcOnw6NvIMOpIG9yaWdpbmFsIGUgcXVlIHZvY8OqIHRlbSBvIHBvZGVyIGRlIGNvbmNlZGVyIG9zIGRpcmVpdG9zIGNvbnRpZG9zIG5lc3RhIGxpY2Vuw6dhLiBWb2PDqiB0YW1iw6ltIGRlY2xhcmEgcXVlIG8gZGVww7NzaXRvIGRlIHN1YSBwdWJsaWNhw6fDo28gbsOjbywgcXVlIHNlamEgZGUgc2V1IGNvbmhlY2ltZW50bywgaW5mcmluZ2UgZGlyZWl0b3MgYXV0b3JhaXMgZGUgbmluZ3XDqW0uCgpDYXNvIGEgc3VhIHB1YmxpY2HDp8OjbyBjb250ZW5oYSBtYXRlcmlhbCBxdWUgdm9jw6ogbsOjbyBwb3NzdWkgYSB0aXR1bGFyaWRhZGUgZG9zIGRpcmVpdG9zIGF1dG9yYWlzLCB2b2PDqiBkZWNsYXJhIHF1ZSBvYnRldmUgYSBwZXJtaXNzw6NvIGlycmVzdHJpdGEgZG8gZGV0ZW50b3IgZG9zIGRpcmVpdG9zIGF1dG9yYWlzIHBhcmEgY29uY2VkZXIgYW8gUmVwb3NpdMOzcmlvIEluc3RpdHVjaW9uYWwgZGEgVUZNRyBvcyBkaXJlaXRvcyBhcHJlc2VudGFkb3MgbmVzdGEgbGljZW7Dp2EsIGUgcXVlIGVzc2UgbWF0ZXJpYWwgZGUgcHJvcHJpZWRhZGUgZGUgdGVyY2Vpcm9zIGVzdMOhIGNsYXJhbWVudGUgaWRlbnRpZmljYWRvIGUgcmVjb25oZWNpZG8gbm8gdGV4dG8gb3Ugbm8gY29udGXDumRvIGRhIHB1YmxpY2HDp8OjbyBvcmEgZGVwb3NpdGFkYS4KCkNBU08gQSBQVUJMSUNBw4fDg08gT1JBIERFUE9TSVRBREEgVEVOSEEgU0lETyBSRVNVTFRBRE8gREUgVU0gUEFUUk9Dw41OSU8gT1UgQVBPSU8gREUgVU1BIEFHw4pOQ0lBIERFIEZPTUVOVE8gT1UgT1VUUk8gT1JHQU5JU01PLCBWT0PDiiBERUNMQVJBIFFVRSBSRVNQRUlUT1UgVE9ET1MgRSBRVUFJU1FVRVIgRElSRUlUT1MgREUgUkVWSVPDg08gQ09NTyBUQU1Cw4lNIEFTIERFTUFJUyBPQlJJR0HDh8OVRVMgRVhJR0lEQVMgUE9SIENPTlRSQVRPIE9VIEFDT1JETy4KCk8gUmVwb3NpdMOzcmlvIEluc3RpdHVjaW9uYWwgZGEgVUZNRyBzZSBjb21wcm9tZXRlIGEgaWRlbnRpZmljYXIgY2xhcmFtZW50ZSBvIHNldSBub21lKHMpIG91IG8ocykgbm9tZXMocykgZG8ocykgZGV0ZW50b3IoZXMpIGRvcyBkaXJlaXRvcyBhdXRvcmFpcyBkYSBwdWJsaWNhw6fDo28sIGUgbsOjbyBmYXLDoSBxdWFscXVlciBhbHRlcmHDp8OjbywgYWzDqW0gZGFxdWVsYXMgY29uY2VkaWRhcyBwb3IgZXN0YSBsaWNlbsOnYS4KRepositório InstitucionalPUBhttps://repositorio.ufmg.br/oaiopendoar:2023-03-29T14:27:15Repositório Institucional da UFMG - Universidade Federal de Minas Gerais (UFMG)false
|