A local limit theorem and loss of rotational symmetry of planar symmetric simple random walk

Beneš, Christian

doi:10.1090/tran/7399

Cited by 3 publications

(8 citation statements)

References 26 publications

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“…Note that as it is remarked in [3], this expression can also be obtained from [11, Theorem 6.1.6] by an explicit calculation of the rate function.…”

Section: Forestsmentioning

confidence: 63%

Classification of scaling limits of uniform quadrangulations with a boundary

2019

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We study non-compact scaling limits of uniform random planar quadrangulations with a boundary when their size tends to infinity. Depending on the asymptotic behavior of the boundary size and the choice of the scaling factor, we observe different limiting metric spaces. Among well-known objects like the Brownian plane or the infinite continuum random tree, we construct two new one-parameter families of metric spaces that appear as scaling limits: the Brownian half-plane with skewness parameter θ and the infinite-volume Brownian disk of perimeter σ. We also obtain various coupling and limit results clarifying the relation between these objects. (GRAAL) and ANR-15-CE40-0013 (Liouville), and of Fondation Simone et Cino Del Duca.Definition 2.1. Let T > 0. The continuum random tree CRT T with volume T is the random rooted real tree (T X , d x , ρ) for the probability distribution that makes the canonical process X of C([0, T ], R) the standard Brownian excursion with duration T .The term "CRT" usually denotes CRT 1 with volume T = 1, in which case X is taken under the law of the normalized Brownian excursion. We simply write CRT instead of CRT 1 .Note the scaling relation, for λ, T > 0:This comes from the fact that, if e T is a Brownian excursion with duration T , then λe T (·/λ 2 ) has same distribution as e λ 2 T .We should also discuss the role of ρ in the above definition. The re-rooting property of CRT T [2, (20)] states, roughly speaking, that if ρ is a random variable with distribution µ X /µ X (1) (the normalized version of the measure defined above), then (T X , d X , ρ ) has same distribution as (T X , d X , ρ). In this sense, the point ρ plays no distinguished role in the construction of CRT T .Brownian map BM T , T > 0. The Brownian map is roughly speaking the metric gluing of the tree coded by a snake driven by a normalized Brownian excursion, on the tree coded by the excursion itself.Definition 2.2. The Brownian map BM T with volume T is the metric space (M X,W , D X,W , ρ) for the probability law that makes X a Brownian excursion of duration T , and W is the snake driven by X.We write BM instead of BM 1 . The scaling properties of Gaussian processes imply easily that for λ > 0, λ · BM T = d BM λ 4 T .Just as for CRT T , the point ρ in BM T should be seen as a random choice according to the normalized measure µ X,W /µ X,W (1), which is known as the re-rooting property of the Brownian map (Theorem 8.1 of [28]). The latter is a crucial property for characterizing the Brownian map, see, e.g., the recent work [35].Brownian disk BD T,σ , σ ∈ (0, ∞), T > 0. The Brownian disk first appears in [7] as limiting metric space along suitable infinite subsequences. Uniqueness of the limit and a concrete description of the metric were obtained in [9].The description is slightly more elaborate than that of the Brownian map. For t ≥ 0, we let X t = inf [0,t] X.Definition 2.3. The Brownian disk BD T,σ with volume T and boundary length σ is the metric space (M X,W , D X,W , ρ) under the probability measures that makes X a f...

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“…Note that as it is remarked in [3], this expression can also be obtained from [11, Theorem 6.1.6] by an explicit calculation of the rate function.…”

Section: Forestsmentioning

confidence: 63%

Classification of scaling limits of uniform quadrangulations with a boundary

2019

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“…A simple consequence of these multi-dimensional results is the loss of symmetry of the random walks on Z d conditioned to be far away from the origin; this loss of symmetry has also been brought out in dimension 2 in [Ben19]. Thus, consider the simple 2-dimensional random walk S n = ∑ n k=1 A k , where A k = (±1, 0) or (0, ±1) with probability 1 4 for each direction.…”

Section: Examples Of Multi-dimensional Convergent Sequencesmentioning

confidence: 99%

Gaussian approximations for random vectors

Méliot¹,

Nikeghbali²

2022

Preprint

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We present several refinements on the fluctuations of sequences of random vectors (with values in the Euclidean space R d ) which converge after normalization to a multidimensional Gaussian distribution. More precisely we refine such results in two directions: first we give conditions under which one can obtain bounds on the speed of convergence to the multidimensional Gaussian distribution, and then we provide a setting in which one can obtain precise moderate or large deviations (in particular we see at which scale the Gaussian approximation for the tails ceases to hold and how the symmetry of the Gaussian tails is then broken). These results extend some of our earlier works obtained for real valued random variables, but they are not simple extensions, as some new phenomena are observed that could not be visible in one dimension. Even for very simple objects such as the symmetric random walk in Z d , we observe a loss of symmetry that we can quantify for walks conditioned to be far away from the origin. Also, unlike the one dimensional case where the Kolmogorov distance is natural, in the multidimensional case there is no more such a canonical distance. We choose to work with the so-called convex distance, and as a consequence the geometry of the Borel measurable sets that we consider shall play an important role (also making the proofs more complicated). We illustrate our results with some examples such as correlated random walks, the characteristic polynomials of random unitary matrices, or pattern countings in random graphs. CONTENTS1. Introduction 1.1. Notational conventions 1.2. Mod-Gaussian convergence 1.3. The one-dimensional case 1.4. Main results and outline of the article Acknowledgement 2. Estimates on the speed of convergence 2.1. Convex distance between probability measures 2.2. Smoothing techniques 2.3. Distance between Fourier transforms 2.4. Berry-Esseen type estimates 3. Large deviation results 3.1. Sum of a Gaussian noise and an independent random variable 3.2. Asymptotics of the probabilities of cones 3.3. Approximation of spherical sectors by unions of cones 4. Examples of multi-dimensional convergent sequences 4.1. First examples 4.2. The method of cumulants 4.3. Sparse dependency graphs 4.4. Empirical measures of Markov chains References

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“…However, this is as far as the expression in (1.2) is valid. Indeed, we showed in Beneš (2019) that in the two-dimensional case, the quantity (1.2) ceases to describe the probability in (1.1) when ∥x∥ 2 is of order greater than n 3/4 . For such x, there are correction terms in the exponent of (1.2) which cause the probability to decay faster, which is unsurprising, since for ∥x∥ 1 > n, P (S (d) (n) = x) = 0.…”

Section: Introductionmentioning

confidence: 97%

“…For such x, there are correction terms in the exponent of (1.2) which cause the probability to decay faster, which is unsurprising, since for ∥x∥ 1 > n, P (S (d) (n) = x) = 0. The number of such correction terms grows with ∥x∥ 2 and tends to infinity as ∥x∥ 1 → n. See Theorem 2.2 and Figure 1 in Beneš (2019) for more details. One interesting consequence of the expression obtained in Beneš (2019) for the probability in (1.1) is that the position at time n of planar simple random walk is approximately (in a sense that is made precise in that paper) rotationally symmetric if one considers points in any disk of radius o(n 3/4 ) but is not approximately rotationally symmetric for points outside of any disk of radius of order greater than n 3/4 .…”

Section: Introductionmentioning

confidence: 98%

“…, x d ) ∈ Z d at time n ∈ Z + has a long history, dating back to over a century ago (see Pólya (1921)). Khintchine (1929) provided a close to complete solution to the problem in dimension 1, and since then, the analysis of the probability in (1.1), both for simple random walk and more general walks has continued to be developed in an important body of work, from Cramér (1938) to more recent results of varying levels of generality (see for instance Borovkov and Mogulskii (1999), Lawler (1991), Lawler and Limic (2010), or Beneš (2019)), some of which rely on the fundamental developments of large deviations theory in Varadhan (1966). See the Appendix in Beneš (2019) for a brief historical description of the evolution of this area of probability theory.…”

Section: Introductionmentioning

confidence: 99%

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On the Large Deviations Rate Function for Symmetric Simple Random Walk in Dimension $d \in \mathbb{Z}_{+}$

Beneš

2024

ALEA

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The large deviations rate function Λ (d) of d-dimensional symmetric simple random walk S (d) (n) describes the log-asymptotic behavior of the fundamental probability P (S (d) (n) = x) in the large deviations regime, that is, when n = O ∥x∥ , but also in the so-called moderate deviations regime, that is, when n 1/2 = o(∥x∥) and ∥x∥ = o(n). In Beneš (2019), we provided precise asymptotics for P (S (d) (n) = x) in dimensions 1 and 2 and, as an immediate consequence, we obtained Λ (d) for d = 1 and d = 2. The techniques developed in that paper do not translate readily to higher dimensions. In the present paper, we show that for d ∈ Z + ,where α = (α 1 , . . . , α d ) satisfies d i=1 |α i | ≤ 1. The quantities a i = a i (α) are not explicit but are given in terms of the solution of a system of quadratic equations. This result represents partial progress towards the generalization of the results in Beneš (2019) to any dimension d ∈ Z + .

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A local limit theorem and loss of rotational symmetry of planar symmetric simple random walk

Cited by 3 publications

References 26 publications

Classification of scaling limits of uniform quadrangulations with a boundary

Classification of scaling limits of uniform quadrangulations with a boundary

Gaussian approximations for random vectors

On the Large Deviations Rate Function for Symmetric Simple Random Walk in Dimension $d \in \mathbb{Z}_{+}$

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