Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome?

Ahlberg, Daniel; Steif, Jeffrey E.

doi:10.1214/16-aihp786

Cited by 7 publications

(16 citation statements)

References 53 publications

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“…FSS at p ∞ remains Gaussian in its raw form (effectively unadorned by dangerous irrelevant variables) [40]. This is well established in φ 4 -theory [28][29][30][31][32][33]69] and implied from early work on the subject [69]. Here we have extended these considerations to φ 3 -theory.…”

Section: Discussionmentioning

confidence: 65%

“…More recently it has been proven that scaling in high dimensions is dependent on boundary conditions at the infinite-volume critical point. While proliferation of the largest clusters is expected for free boundary conditions (FBCs), where they have fractal dimension D = 4 [20], rigorous proofs have established that, under certain conditions, D = 2d/3 for other boundary conditions, such as periodic boundary conditions (PBCs), d being the dimension of the underlying lattice [24][25][26][27][28]. This is known as random-graph asymptotics because, with V = L d , it is similar to the behavior of the largest critical cluster on the complete graph of V sites [16,26].…”

Section: Introductionmentioning

confidence: 99%

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Universal finite-size scaling for percolation theory in high dimensions

Kenna¹,

Berche²

2017

J. Phys. A: Math. Theor.

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We present a unifying, consistent, finite-size-scaling picture for percolation theory bringing it into the framework of a general, renormalization-group-based, scaling scheme for systems above their upper critical dimensions d c . Behaviour at the critical point is non-universal in d > d c = 6 dimensions. Proliferation of the largest clusters, with fractal dimension 4, is associated with the breakdown of hyperscaling there when free boundary conditions are used. But when the boundary conditions are periodic, the maximal clusters have dimension D = 2d/3, and obey random-graph asymptotics. Universality is instead manifest at the pseudocritical point, where the failure of hyperscaling in its traditional form is universally associated with random-graph-type asymptotics for critical cluster sizes, independent of boundary conditions.

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Section: Discussionmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 99%

Universal finite-size scaling for percolation theory in high dimensions

Kenna¹,

Berche²

2017

J. Phys. A: Math. Theor.

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“…We remark that the width of the critical window cannot decay faster than order 1 / √ n due to the well-known fact that no sequence of monotone Boolean functions may have a smaller threshold window; see e.g. [3]. We mention that in parallel work, Duminil-Copin, Raoufi and Tassion [9] present yet another proof of the Bollobás-Riordan theorem.…”

Section: Description Of Resultsmentioning

confidence: 87%

“…A more modern approach to threshold phenomena comes from randomized algorithms via the OSSS inequality [18]. That randomized algorithms can be used to study threshold phenomena has previously been observed by Gady Kozma (see the appendix of [3]) and in recent work by Duminil-Copin, Raoufi and Tassion [9,10]. Randomized algorithms are also connected to noise sensitivity via the Schramm-Steif revealment theorem [21].…”

Section: Description Of Resultsmentioning

confidence: 99%

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Noise sensitivity and Voronoi percolation

Ahlberg¹,

Baldasso²

2018

Electron. J. Probab.

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In this paper we study noise sensitivity and threshold phenomena for Poisson Voronoi percolation on R 2 . In the setting of Boolean functions, both threshold phenomena and noise sensitivity can be understood via the study of randomized algorithms. Together with a simple discretization argument, such techniques apply also to the continuum setting. Via the study of a suitable algorithm we show that boxcrossing events in Voronoi percolation are noise sensitive and present a threshold phenomenon with polynomial window. We also study the effect of other kinds of perturbations, and emphasize the fact that the techniques we use apply for a broad range of models.

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Volatility of Boolean functions

Jonasson

Steif

2016

Stochastic Processes and their Applications

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We study the volatility of the output of a Boolean function when the input bits undergo a natural dynamics. For n = 1, 2, . . ., let f n : {0, 1} mn → {0, 1} be a Boolean function and X (n) (t) = (X 1 (t), . . . , X mn (t)) t∈[0,∞) be a vector of i.i.d. stationary continuous time Markov chains on {0, 1} that jump from 0 to 1 with rate p n ∈ [0, 1] and from 1 to 0 with rate q n = 1 − p n . Our object of study will be C n which is the number of state changes of f n (X (n) (t)) as a function of t during [0, 1]. We say that the family {f n } n≥1 is volatile if C n → ∞ in distribution as n → ∞ and say that {f n } n≥1 is tame if {C n } n≥1 is tight. We study these concepts in and of themselves as well as investigate their relationship with the recent notions of noise sensitivity and noise stability. In addition, we study the question of lameness which means that P(C n = 0) → 1 as n → ∞. Finally, we investigate these properties for the majority function, iterated 3-majority, the AND/OR function on the binary tree and percolation on certain trees in various regimes. lim n→∞ P(f n (X (n) (0)) = 1)P(f n (X (n) (0)) = 0) = 0 and nondegenerate with respect to {p n } if for some δ > 0,for all n. (Note that a sequence can of course be neither degenerate nor nondegenerate although it will always have a subsequence which is either one or the other.) The first concept we give captures the notion that it is unlikely that there is any change of state. Definition 1.1 We say that {f n } n≥1 is lame with respect to {p n } if lim n→∞ P(C n ) = 0) = 1.The first relatively easy proposition says that a necessary condition for lameness is that the sequence is degenerate. Proposition 1.2 Let {f n } n≥1 be a sequence of Boolean functions and {p n } be a sequence in [0, 1]. If {f n } n≥1 is lame with respect to {p n }, then it is degenerate with respect to {p n }.The following two definitions will be central to the paper.

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Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome?

Cited by 7 publications

References 53 publications

Universal finite-size scaling for percolation theory in high dimensions

Universal finite-size scaling for percolation theory in high dimensions

Noise sensitivity and Voronoi percolation

Volatility of Boolean functions

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