Large Deviations of the Range of the Planar Random Walk on the Scale of the Mean
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Large Deviations of the Range of the Planar Random Walk on the Scale of the Mean Jingjia Liu1 · Quirin Vogel2 Received: 28 April 2020 / Revised: 24 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We prove an upper large deviation bound on the scale of the mean for a symmetric random walk in the plane satisfying certain moment conditions. This result complements the study by Phetpradap for the random walk range, which is restricted to dimension three and higher, and of van den Berg, Bolthausen and den Hollander, for the volume of the Wiener sausage. Keywords Large deviations · Random walk range · Planar random walk Mathematics Subject Classification (2010) Primary 60G50; Secondary 60F10
1 Introduction The range of the random walk is a topic which n has been studied for more than 60 X i is defined to be the sum of i.i.d. years. For our purpose, a random walk Sn = i=1 random variables on Zd , where X i has mean zero. The range Rn of a random walk is then defined as Rn = #{x ∈ Zd : ∃ k ∈ {0, . . . n} with Sk = x}. Previous works, for example [8,20], examined the mean and the variance of Rn . It was proven that the mean range of a fairly general random walk (with identity covariance) is given by
B
Quirin Vogel [email protected] Jingjia Liu [email protected]
1
Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstraße 62, 48149 Münster, Germany
2
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
123
Journal of Theoretical Probability
E[Rn ] ∼
⎧ 1/2 8n ⎪ ⎨ π 2π n ⎪ log n
⎩
κd n
if d = 1, if d = 2,
(1.1)
if d ≥ 3,
with κd = P(Si = 0 for all i ≥ 1), see [7, Equation 5.3.39].1 Estimations of error terms as well as asymptotics for the variance of Rn are also available from the aforementioned references. As proof techniques developed, large deviation results on the scale n d/(d+2) were obtained in [13]. Central limit theorems were given in [19,21]. Laws of the iterated logarithm can be found in [5]. For a good overview of these classical results, we refer the reader to [7], where more precise statements are presented. In recent years, the understanding of different properties of the range of the random walk had been refined. We present a (very incomplete) selection of these results. In [4], moderate deviations of the renormalized range Rn − E[Rn ] were studied. Uchiyama gave in [27] an asymptotic expansion of the expectation of the range of a random walk bridge, which holds uniformly in a large set of possible end points of the bridge. In a sequence of papers [1–3], the capacity of the range of the random walk was analyzed, with a focus on precise results in high dimensions. A strong law of large numbers type result for the boundary of the range of the random walk was obtained in [9] for both transient and recurrent random walks. Additionally, the range of a planar random walk conditioned on never hitting the origin was studied in [16]. The study of the range of random walks has a lot of important applications. For i
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