The Cut-off Phenomenon for Brownian Motions on Compact Symmetric Spaces

Méliot, Pierre-Loïc

doi:10.1007/s11118-013-9356-7

Cited by 27 publications

(27 citation statements)

References 27 publications

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“…At this point we refrain from giving a full account on the mathematical literature on the cutoff phenomenon and refer to the overview article [41] and the introduction of [16]. Standard references in the mathematics literature on the cutoff phenomenon for discrete time and space include [1,3,4,9,15,18,32,33,40,42,[74][75][76][77][78]80,103,109]. As introductory texts on the cutoff phenomenon in discrete time and space we recommend [68] and Chapter 18 in the monograph [78].…”

Section: Introductionmentioning

confidence: 99%

Cutoff Thermalization for Ornstein–Uhlenbeck Systems with Small Lévy Noise in the Wasserstein Distance

2021

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This article establishes cutoff thermalization (also known as the cutoff phenomenon) for a class of generalized Ornstein–Uhlenbeck systems $$(X^\varepsilon _t(x))_{t\geqslant 0}$$ ( X t ε ( x ) ) t ⩾ 0 with $$\varepsilon $$ ε -small additive Lévy noise and initial value x. The driving noise processes include Brownian motion, $$\alpha $$ α -stable Lévy flights, finite intensity compound Poisson processes, and red noises, and may be highly degenerate. Window cutoff thermalization is shown under mild generic assumptions; that is, we see an asymptotically sharp $$\infty /0$$ ∞ / 0 -collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure $$\mu ^\varepsilon $$ μ ε along a time window centered on a precise $$\varepsilon $$ ε -dependent time scale $$\mathfrak {t}_\varepsilon $$ t ε . In many interesting situations such as reversible (Lévy) diffusions it is possible to prove the existence of an explicit, universal, deterministic cutoff thermalization profile. That is, for generic initial data x we obtain the stronger result $$\mathcal {W}_p(X^\varepsilon _{t_\varepsilon + r}(x), \mu ^\varepsilon ) \cdot \varepsilon ^{-1} \rightarrow K\cdot e^{-q r}$$ W p ( X t ε + r ε ( x ) , μ ε ) · ε - 1 → K · e - q r for any $$r\in \mathbb {R}$$ r ∈ R as $$\varepsilon \rightarrow 0$$ ε → 0 for some spectral constants $$K, q>0$$ K , q > 0 and any $$p\geqslant 1$$ p ⩾ 1 whenever the distance is finite. The existence of this limit is characterized by the absence of non-normal growth patterns in terms of an orthogonality condition on a computable family of generalized eigenvectors of $$\mathcal {Q}$$ Q . Precise error bounds are given. Using these results, this article provides a complete discussion of the cutoff phenomenon for the classical linear oscillator with friction subject to $$\varepsilon $$ ε -small Brownian motion or $$\alpha $$ α -stable Lévy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature.

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Section: Introductionmentioning

confidence: 99%

Cutoff Thermalization for Ornstein–Uhlenbeck Systems with Small Lévy Noise in the Wasserstein Distance

2021

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“…Now we will specify (16) to all compact classical groups: to do this, we only need to know the dual space G, and the numbers c λ and d λ for every λ ∈ G. This can be done thanks to their root systems. Most definitions and results are borrowed from [4], and can also be recovered with much clarity from Sections 2.2 and 2.3 in [40]. Definition 3.2.…”

Section: Character Decomposition Of the Partition Functionmentioning

confidence: 99%

Large N limit of Yang-Mills partition function and Wilson loops on compact surfaces

Dahlqvist¹,

Lemoine²

2022

Preprint

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We compute the large N limit of several objects related to the two-dimensional Euclidean Yang-Mills measure on compact connected orientable surfaces Σ of genus g ≥ 1 with a structure group taken among the classical groups of order N . We first generalise to all classical groups the convergence of partitions functions obtained by the second author in [29], then we apply this result to show that all Wilson loop observables for loops included within a topological disc of Σ converge towards the limit of the Wilson loop where the latter disc is embedded in the plane in place of Σ. This solves a conjecture of B. Hall [25] for closed surfaces. Besides, using similar arguments, we show that Wilson loops vanish for all non-contractible simple loops.

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“…The literature on those topics is nowadays quite substantial. We for instance refer to the early works by Eugene Dynkin [14,15] and Paul & Marie-Paule Malliavin [24] or more recent presentations like [25] and the book [16] (see in particular Chapter 8: Riemannian submersions and Symmetric spaces). In some sense, Theorem 2.1 provides a more pedestrian approach: We work in a specific choice of coordinates within the algebra of complex matrices and describe the G n,k and V n,k Brownian motions in those coordinates taking advantage of the additional structure given by the matrix multiplication.…”

Section: Remark 22mentioning

confidence: 99%

Asymptotic windings of the block determinants of a unitary Brownian motion and related diffusions

Baudoin

Wang²

2021

Electron. J. Probab.

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We study several matrix diffusion processes constructed from a unitary Brownian motion. In particular, we use the Stiefel fibration to lift the Brownian motion of the complex Grassmannian to the complex Stiefel manifold and deduce a skew-product decomposition of the Stiefel Brownian motion. As an application, we prove asymptotic laws for the determinants of the block entries of the unitary Brownian motion.

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The Cut-off Phenomenon for Brownian Motions on Compact Symmetric Spaces

Cited by 27 publications

References 27 publications

Cutoff Thermalization for Ornstein–Uhlenbeck Systems with Small Lévy Noise in the Wasserstein Distance

Cutoff Thermalization for Ornstein–Uhlenbeck Systems with Small Lévy Noise in the Wasserstein Distance

Large N limit of Yang-Mills partition function and Wilson loops on compact surfaces

Asymptotic windings of the block determinants of a unitary Brownian motion and related diffusions

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