{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:39:50Z","timestamp":1760146790296,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,11]],"date-time":"2024-12-11T00:00:00Z","timestamp":1733875200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Between 2007 and 2018, we collaborated extensively with P\u00e1l R\u00e9v\u00e9sz and Mikl\u00f3s Cs\u00f6rg\u0151 on many of the problems discussed in this paper. Over the past six years, we have continued to explore these issues, and here, we present some of the most intriguing open questions in these areas. This paper compiles key results from a dozen of our previous works, providing the necessary background to frame these compelling unresolved questions.<\/jats:p>","DOI":"10.3390\/e26121082","type":"journal-article","created":{"date-parts":[[2024,12,11]],"date-time":"2024-12-11T12:58:49Z","timestamp":1733921929000},"page":"1082","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Some Open Questions About the Anisotropic Random Walks"],"prefix":"10.3390","volume":"26","author":[{"given":"Endre","family":"Cs\u00e1ki","sequence":"first","affiliation":[{"name":"Alfr\u00e9d R\u00e9nyi Institute of Mathematics, P.O. Box 127, H-1364 Budapest, Hungary"}]},{"given":"Ant\u00f3nia","family":"F\u00f6ldes","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Staten Island, The City University of New York, 2800 Victory Blvd., Staten Island, NY 10314, USA"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Silver, H., Shuler, K.E., and Lindenberg, K. (1977). Two-dimensional anisotropic random walks. Statistical Mechanics and Statistical Methods in Theory and Application (Proc. Sympos., Univ. Rochester, Rochester, N.Y., 1976), Plenum.","DOI":"10.1007\/978-1-4613-4166-6_21"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/0378-4371(79)90042-6","article-title":"Random walks on sparsely periodic and random lattices I","volume":"95","author":"Shuler","year":"1979","journal-title":"Phys. A"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1007\/BF01011166","article-title":"Random Walks on Periodic and Random Lattices. II. Random Walk Properties via Generating Function Techniques","volume":"21","author":"Seshadri","year":"1979","journal-title":"J. Stat. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/BF01011741","article-title":"Random walks on a lattice","volume":"27","author":"Westcott","year":"1982","journal-title":"J. Stat. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1007\/BF01013444","article-title":"On the asymptotic behavior of random walks on an anisotropic lattice","volume":"27","author":"Heyde","year":"1982","journal-title":"J. Stat. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Heyde, C.C. (1993). Asymptotics for two-dimensional anisotropic random walks. Stochastic Processes, Springer.","DOI":"10.1007\/978-1-4615-7909-0_15"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1007\/BF01010534","article-title":"Asymptotic properties of multistate random walks. I. Theory","volume":"40","author":"Roerdink","year":"1985","journal-title":"J. Stat. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1007\/BF02186749","article-title":"On three conjectures by K. Shuler","volume":"75","year":"1994","journal-title":"J. Stat. Phys."},{"key":"ref_9","first-page":"137","article-title":"Some problems concerning the structure of random walk paths","volume":"11","author":"Taylor","year":"1960","journal-title":"Acta Math. Acad. Sci. Hung."},{"key":"ref_10","unstructured":"Dvoretzky, A., and Erd\u0151s, P. (August, January 31). Some problems on random walk in space. Proceedings of the Second Berkeley Symposium, Berkeley, CA, USA."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"R\u00e9v\u00e9sz, P. (2005). Random Walk in Random and Non-Random Environments, World Scientific. [2nd ed.].","DOI":"10.1142\/9789812703361"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1016\/0378-4371(86)90060-9","article-title":"Some properties of a random walk on a comb structure","volume":"134","author":"Weiss","year":"1986","journal-title":"Phys. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1184","DOI":"10.1214\/EJP.v11-377","article-title":"Asymptotic behaviour of the simple random walk on the 2-dimensional comb","volume":"11","author":"Bertacchi","year":"2006","journal-title":"Electron. J. Probab."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1017\/S1446788700008144","article-title":"Uniform asymptotic estimates of transition probabilities on combs","volume":"75","author":"Bertacchi","year":"2003","journal-title":"J. Aust. Math. Soc."},{"key":"ref_15","first-page":"2371","article-title":"Strong limit theorems for a simple random walk on the 2-dimensional comb","volume":"14","year":"2009","journal-title":"Electron. J. Probab."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1290","DOI":"10.1016\/j.spa.2011.01.009","article-title":"On the local time of random walk on the 2-dimensional comb","volume":"121","year":"2011","journal-title":"Stoch. Process. Appl."},{"key":"ref_17","first-page":"55","article-title":"Some results and problems for anisotropic random walk on the plane","volume":"76","year":"2013","journal-title":"Fields Inst. Commun."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1017\/S0305004100033879","article-title":"Random walks and electric currents in networks","volume":"55","year":"1959","journal-title":"Proc. Camb. Phil. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s10998-013-7408-9","article-title":"Strong limit theorems for anisotropic random walks on Z2","volume":"67","year":"2013","journal-title":"Periodica Math. Hung."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1007\/BF01012611","article-title":"The asymptotic behavior of a random walk on a dual-medium lattice","volume":"28","author":"Heyde","year":"1982","journal-title":"J. Stat. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1239\/jap\/1032192563","article-title":"Diffusion approximation for random walks on anisotropic lattices","volume":"35","year":"1998","journal-title":"J. Appl. Probab."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"300","DOI":"10.2307\/3213403","article-title":"Oscillating Brownian motion","volume":"15","author":"Keilson","year":"1978","journal-title":"J. Appl. Probab."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2879","DOI":"10.1007\/s10959-021-01142-0","article-title":"Strong approximation of the anisotropic random walk revisited","volume":"35","year":"2022","journal-title":"J. Theor. Probab."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2233","DOI":"10.1007\/s10959-019-00938-5","article-title":"Random walks on comb-type subsets of Z2","volume":"33","year":"2020","journal-title":"J. Theor. Probab."},{"key":"ref_25","first-page":"29","article-title":"Random walk on half-plane half-comb structure","volume":"39","year":"2012","journal-title":"Ann. Math. Inform."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1247","DOI":"10.1007\/s10959-020-01065-2","article-title":"On the local time of the Half-Plane Half-Comb walk","volume":"35","year":"2022","journal-title":"J. Theor. Probab."},{"key":"ref_27","first-page":"29","article-title":"Random walks on the two-dimensional K-comb lattice","volume":"29","year":"2023","journal-title":"Math. Pannonica"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/S0304-4149(01)00079-5","article-title":"Moderate deviations for Markovian occupation time","volume":"94","author":"Chen","year":"2001","journal-title":"Stoch. Process. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s10998-010-3165-1","article-title":"We like to walk on the comb","volume":"61","year":"2010","journal-title":"Period. Math. Hung."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1016\/0012-365X(86)90107-X","article-title":"Natural spanning trees of Zd are recurrent","volume":"61","author":"Gerl","year":"1986","journal-title":"Discret. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1654","DOI":"10.1007\/s10959-023-01297-y","article-title":"On the local time of the anisotropic walk on Z2","volume":"37","year":"2024","journal-title":"J. Theor. Probab."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1214\/ECP.v18-2336","article-title":"A local limit theorem for random walks in balanced environments","volume":"18","author":"Stenlund","year":"2013","journal-title":"Electron. Commun. Probab."},{"key":"ref_33","first-page":"373","article-title":"Number of distinct sites visited by a random walk with internal states","volume":"150","year":"2010","journal-title":"Probab. Theory Relat. Fields"},{"key":"ref_34","first-page":"1","article-title":"The range of a random walk on a comb","volume":"20","author":"Pach","year":"2013","journal-title":"Electron. J. Comb."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1007\/BF00539114","article-title":"An invariance principle for the law of iterated logarithm","volume":"4","author":"Strassen","year":"1964","journal-title":"Z. Wahrscheinlichkeitstheorie Verw. Geb."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"444","DOI":"10.1090\/S0002-9947-1957-0084222-7","article-title":"On occupation times for Markoff processes","volume":"84","author":"Darling","year":"1957","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1016\/S0246-0203(98)80027-7","article-title":"Functional laws of the iterated logarithm for local times of recurrent random walks on Z2","volume":"34","author":"Rosen","year":"1998","journal-title":"Ann. Inst. Henri Poincar\u00e9 Probab. Stat."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1153","DOI":"10.1090\/S0002-9947-97-01717-0","article-title":"Strassen theorems for a class of iterated processes","volume":"349","year":"1997","journal-title":"Trans. Amer. Math. 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