Random-Turn Hex and Other Selection Games

Peres, Yuval; Schramm, Oded; Sheffield⋆, Scott; Wilson, David B.

doi:10.1080/00029890.2007.11920428

Cited by 40 publications

(36 citation statements)

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“…Since the probability of points not too close to the boundary being pivotal is about R −5/4+o(1) (this is the 4 arm event) and for a monotone function f ,f ({i}) is the probability that x i is pivotal, the case k = 1 in Theorem 1.8 implies that the revealment is at least R −1/2+o (1) . As pointed out in Peres, Schramm, Sheffield and Wilson [25], this can also be obtained using an inequality of O'Donnell and Servedio. Theorems 1.8 and 4.1 immediately give…”

Section: Simply Connected Casementioning

confidence: 93%

Quantitative noise sensitivity and exceptional times for percolation

2010

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One goal of this paper is to prove that dynamical critical site percolation on the planar triangular lattice has exceptional times at which percolation occurs. In doing so, new quantitative noise sensitivity results for percolation are obtained. The latter is based on a novel method for controlling the "level k" Fourier coefficients via the construction of a randomized algorithm which looks at random bits, outputs the value of a particular function but looks at any fixed input bit with low probability. We also obtain upper and lower bounds on the Hausdorff dimension of the set of percolating times. We then study the problem of exceptional times for certain "k-arm" events on wedges and cones. As a corollary of this analysis, we prove, among other things, that there are no times at which there are two infinite "white" clusters, obtain an upper bound on the Hausdorff dimension of the set of times at which there are both an infinite white cluster and an infinite black cluster and prove that for dynamical critical bond percolation on the square grid there are no exceptional times at which 3 disjoint infinite clusters are present.

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Section: Simply Connected Casementioning

confidence: 93%

Quantitative noise sensitivity and exceptional times for percolation

2010

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“…However, the additional dimension complicates the visualization process, thus rendering the analysis more difficult. In addition, it would be very interesting to extend these techniques to partizan games, 3 like chess and checkers ͑we have made preliminary progress in generalizing this methodology to very simple partizan games͒, or even to consider games with intrinsic randomness, such as the recent analysis of hex with coin flips, 22 or even backgammon.…”

Section: E Application To Other Gamesmentioning

confidence: 99%

Nonlinear dynamics in combinatorial games: Renormalizing Chomp

Friedman

Landsberg

2007

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We develop a new approach to combinatorial games that reveals connections between such games and some of the central ideas of nonlinear dynamics: scaling behaviors, complex dynamics and chaos, universality, and aggregation processes. We take as our model system the combinatorial game Chomp, which is one of the simplest in a class of "unsolved" combinatorial games that includes Chess, Checkers, and Go. We discover that the game possesses an underlying geometric structure that "grows" ͑reminiscent of crystal growth͒, and show how this growth can be analyzed using a renormalization procedure adapted from physics. In effect, this methodology allows one to transform a combinatorial game like Chomp into a type of dynamical system. Not only does this provide powerful insights into the game of Chomp ͑yielding a complete probabilistic description of optimal play in Chomp and an answer to a longstanding question about the nature of the winning opening move͒, but more generally, it offers a mathematical framework for exploring this unexpected relationship between combinatorial games and modern dynamical systems theory. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2725717͔Combinatorial games, which include Chess, Go, Checkers, Chomp, Dots-and-Boxes, and Nim, have both captivated and challenged mathematicians, computer scientists, and players alike. These are two-player games with no randomness (e.g., no rolling of dice or dealing of cards) and no hidden information (unlike poker, for instance). Apart from their obvious entertainment value, combinatorial games have long been the subject of much serious study owing to the deep mathematical questions they raise in areas such as computational complexity, graph theory, and surreal numbers. 1-3 Using the game of Chomp as a prototype, we report here on a new geometrical approach which unveils unexpected parallels between combinatorial games and some of the central ideas of dynamical systems theory, most notably notions of scaling, renormalization, universality, and chaotic attractors. We show in particular that the game of Chomp can behave in ways analogous to a chaotic dynamical system. This insight and subsequent analysis allows us not only to answer a number of open questions about the game of Chomp, but also provides a new perspective on complex combinatorial games more generally and offers a mathematical framework for understanding this connection between combinatorial games and dynamical systems. Our central finding is that underlying the game of Chomp is a geometric structure that encodes essential information about the game, and that this structure exhibits a type of spatial scale invariance: Loosely speaking, the geometry of "small" and "large" winning positions in the game look the same, after rescaling. We demonstrate that the geometries on different spatial scales can be related to one another via a set of recursion operators, which in turn allows us to recast the game as a type of dynamical system. We then analyze this "dynamical system" using tools and conce...

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“…Originally, we were motivated not by the infinity Laplacian but by random turn Hex [22] and its generalizations, which led us to consider the tug-of-war game. As k−1 i=0 f (x i ).…”

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confidence: 99%

Tug-of-war and the infinity Laplacian

Peres

Schramm

Sheffield⋆

et al. 2008

J. Amer. Math. Soc.

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429

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We prove that every bounded Lipschitz function F on a subset Y of a length space X admits a tautest extension to X, i.e., a unique Lipschitz extension u for which Lip_U u = Lip_{boundary of U} u for all open subsets U of X that do not intersect Y. This was previously known only for bounded domains R^n, in which case u is infinity harmonic, that is, a viscosity solution to Delta_infty u = 0. We also prove the first general uniqueness results for Delta_infty u = g on bounded subsets of R^n (when g is uniformly continuous and bounded away from zero), and analogous results for bounded length spaces. The proofs rely on a new game-theoretic description of u. Let u^epsilon(x) be the value of the following two-player zero-sum game, called tug-of-war: fix x_0=x \in X minus Y. At the kth turn, the players toss a coin and the winner chooses an x_k with d(x_k, x_{k-1})< epsilon. The game ends when x_k is in Y, and player one's payoff is F(x_k) - (epsilon^2/2) sum_{i=0}^{k-1} g(x_i) We show that the u^\epsilon converge uniformly to u as epsilon tends to zero. Even for bounded domains in R^n, the game theoretic description of infinity-harmonic functions yields new intuition and estimates; for instance, we prove power law bounds for infinity-harmonic functions in the unit disk with boundary values supported in a delta-neighborhood of a Cantor set on the unit circle.Comment: 44 pages, 4 figure

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Random-Turn Hex and Other Selection Games

Cited by 40 publications

References 14 publications

Quantitative noise sensitivity and exceptional times for percolation

Quantitative noise sensitivity and exceptional times for percolation

Nonlinear dynamics in combinatorial games: Renormalizing Chomp

Tug-of-war and the infinity Laplacian

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