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Bibliographic Details
Main Author:	Gu, Zhuohan
Format:	Preprint
Published:	2024
Subjects:	Probability
Online Access:	https://arxiv.org/abs/2411.09988
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_version_	1866910699565875200
author	Gu, Zhuohan
author_facet	Gu, Zhuohan
contents	In this paper, we make a few random explorations that relate directly to the items mentioned in the title. We define transient chains and recurrent chains with "killing", the Green's function, the Laplacian operator, and harmonic functions. We then introduce the loop-erased random walk (LERW) and its relationship with the uniform spanning tree (UST). We finish by introducing loop measures and soups and defining the "growing loop" model.
format	Preprint
id	arxiv_https___arxiv_org_abs_2411_09988
institution	arXiv
publishDate	2024
record_format	arxiv
spellingShingle	A Study in Markov Chains, Loop-Erased Random Walk and Loop Soups Gu, Zhuohan Probability In this paper, we make a few random explorations that relate directly to the items mentioned in the title. We define transient chains and recurrent chains with "killing", the Green's function, the Laplacian operator, and harmonic functions. We then introduce the loop-erased random walk (LERW) and its relationship with the uniform spanning tree (UST). We finish by introducing loop measures and soups and defining the "growing loop" model.
title	A Study in Markov Chains, Loop-Erased Random Walk and Loop Soups
topic	Probability
url	https://arxiv.org/abs/2411.09988

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// Anecdotal testing Today and Yesterday seems to indicate destroy() // is called by the garbage collector and everything is good. // Something to keep in mind though. return 0; } /** * Write function that is called when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ public function write($sessId, $data): bool { if ($this->writesDisabled) { return true; } return $this->saveSession($sessId, $data); } /** * A function that is called internally when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ abstract protected function saveSession($sessId, $data): bool; }

* * @return void */ public function writeClose() { // The assumption is that we're using PHP's ext/session. // session_write_close() will actually overwrite $_SESSION with an // empty array on completion -- which leads to a mismatch between what // is in the storage object and $_SESSION. To get around this, we // temporarily reset $_SESSION to an array, and then re-link it to // the storage object. // // Additionally, while you _can_ write to $_SESSION following a // session_write_close() operation, no changes made to it will be // flushed to the session handler. As such, we now mark the storage // object isImmutable. $storage = $this->getStorage(); if (! $storage->isImmutable()) { $_SESSION = $storage->toArray(true); session_write_close(); $storage->fromArray($_SESSION); $storage->markImmutable(); } } /** * Attempt to set the session name * * If the session has already been started, or if the name provided fails * validation, an exception will be raised. * * @param string $name * @return SessionManager * @throws Exception\InvalidArgumentException */ public function setName($name) { if ($this->sessionExists()) { throw new Exception\InvalidArgumentException( 'Cannot set session name after a session has already started'

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