# Mathematics > Probability

arXiv:1311.7439 (math)

# Title:Excited random walk with periodic cookies

Abstract: In this paper we consider an excited random walk on $\mathbb{Z}$ in identically piled periodic environment. This is a discrete time process on $\mathbb{Z}$ defined by parameters $(p_1,\dots p_M) \in [0,1]^M$ for some positive integer $M$, where the walker upon the $i$-th visit to $z \in \mathbb{Z}$ moves to $z+1$ with probability $p_{i\pmod M}$, and moves to $z-1$ with probability $1-p_{i \pmod M}$. We give an explicit formula in terms of the parameters $(p_1,\dots,p_M)$ which determines whether the walk is recurrent, transient to the left, or transient to the right. In particular, in the case that $\frac{1}{M}\sum_{i=1}^{M}p_{i}=\frac {1}{2}$ all behaviors are possible, and may depend on the order of the $p_i$. Our framework allows us to reprove some known results on ERW with no additional effort.
 Comments: 31 pages, no figures. Incorporating several corrections Subjects: Probability (math.PR) Journal reference: Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques 2016, Vol. 52, No. 3, 1023-1049 DOI: 10.1214/15-AIHP669 Cite as: arXiv:1311.7439 [math.PR] (or arXiv:1311.7439v2 [math.PR] for this version)

## Submission history

From: Tal Orenshtein [view email]
[v1] Thu, 28 Nov 2013 22:59:38 UTC (30 KB)
[v2] Sat, 20 Dec 2014 09:30:49 UTC (30 KB)
{"ft_lang_label":"__label__en","ft_lang_prob":0.7384488,"math_prob":0.98639053,"size":1157,"snap":"2020-24-2020-29","text_gpt3_token_len":348,"char_repetition_ratio":0.08326106,"word_repetition_ratio":0.0,"special_character_ratio":0.31633535,"punctuation_ratio":0.11764706,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9932928,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-05-31T00:46:07Z\",\"WARC-Record-ID\":\"<urn:uuid:1bfcc31d-2ab1-41a0-8083-7377f310e73a>\",\"Content-Length\":\"30521\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:55ca9018-c500-41d3-a5d2-6588e7d3a7be>\",\"WARC-Concurrent-To\":\"<urn:uuid:034c8027-a4a8-4693-8863-532b26163f26>\",\"WARC-IP-Address\":\"128.84.21.199\",\"WARC-Target-URI\":\"https://arxiv.org/abs/1311.7439\",\"WARC-Payload-Digest\":\"sha1:5D5JJ2TZ366OIWYITEZIC7DQR5QOKAKU\",\"WARC-Block-Digest\":\"sha1:PCPXMG3PCHKAS4X4G3CX5WGST3ITVSE4\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347410535.45_warc_CC-MAIN-20200530231809-20200531021809-00301.warc.gz\"}"}