Convergence in Law of the Maximum of the Two‐Dimensional Discrete Gaussian Free Field

Bramson, Maury; Ding, Jian; Zeitouni, Ofer

doi:10.1002/cpa.21621

Cited by 132 publications

(266 citation statements)

References 26 publications

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“…A key object for understanding their behavior is the derivative martingale studied here. Similar results have then been obtained for logarithmically correlated Gaussian fields such as the two-dimensional Gaussian Free Field: construction of the derivative martingale [37,38] and study of the extremes of these fields [62,27,36,20,21,19].…”

Section: Introductionsupporting

confidence: 68%

1-stable fluctuations in branching Brownian motion at critical temperature I: The derivative martingale

Maillard¹,

Pain²

2019

Ann. Probab.

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Let (Z t ) t≥0 denote the derivative martingale of branching Brownian motion, i.e. the derivative with respect to the inverse temperature of the normalized partition function at critical temperature. A well-known result by Lalley and Sellke [Ann. Probab., 15(3):1052-1061, 1987 says that this martingale converges almost surely to a limit Z ∞ , positive on the event of survival. In this paper, our concern is the fluctuations of the derivative martingale around its limit. A corollary of our results is the following convergence, confirming and strengthening a conjecture by Mueller and Munier [Phys. Rev. E, 90:042143, 2014]:where S is a spectrally positive 1-stable Lévy process independent of Z ∞ . In a first part of the paper, a relatively short proof of (a slightly stronger form of) this convergence is given based on the functional equation satisfied by the characteristic function of Z ∞ together with tail asymptotics of this random variable. We then set up more elaborate arguments which yield a more thorough understanding of the trajectories of the particles contributing to the fluctuations. In this way, we can upgrade our convergence result to functional convergence. This approach also sets the ground for a follow-up paper, where we study the fluctuations of more general functionals including the renormalized critical additive martingale.All proofs in this paper are given under the hypothesis E L(log L) 3 < ∞, where the random variable L follows the offspring distribution of the branching Brownian motion. We believe this hypothesis to be optimal.

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Section: Introductionsupporting

confidence: 68%

1-stable fluctuations in branching Brownian motion at critical temperature I: The derivative martingale

Maillard¹,

Pain²

2019

Ann. Probab.

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“…Log-correlated Gaussian fields are known to share many properties with BBM. For example, the distribution of the maximum [8,14,42], the overlap of extremal points [3,4,7], and their limiting Gibbs measures [48,51] behave similarly. It is therefore widely believed that their limiting extremal processes exist and should also exhibit the structure of Condition (SDP).…”

Section: If G Satisfies (21) Below Then (Sus) and (Sdp) Are Equivalmentioning

confidence: 81%

Freezing and Decorated Poisson Point Processes

Subag

Zeitouni

2015

Commun. Math. Phys.

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The limiting extremal processes of the branching Brownian motion (BBM), the two-speed BBM, and the branching random walk are known to be randomly shifted decorated Poisson point processes (SDPPP). In the proofs of those results, the Laplace functional of the limiting extremal process is shown to satisfy L θ y f = g y − τ f for any nonzero, nonnegative, compactly supported, continuous function f , where θ y is the shift operator, τ f is a real number that depends on f , and g is a real function that is independent of f . We show that, under some assumptions, this property characterizes the structure of SDPPP. Moreover, when it holds, we show that g has to be a convolution of the Gumbel distribution with some measure.The above property of the Laplace functional is closely related to a 'freezing phenomenon' that is expected to occur in a wide class of log-correlated fields, and which has played an important role in the analysis of various models. Our results shed light on this intriguing phenomenon and provide a natural tool for proving an SDPPP structure in these and other models.

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“…The most interesting and most subtle case is the critical one (d = 2 for the gradient model, d = 4 for the membrane model). For the gradient model, in a series of papers [BDG01, BDZ11, BZ12, BDZ16] it was shown that M ∇ N − m ∇ N converges in distribution to a randomly shifted Gumbel variable, where m ∇ N = 2 π log N − 3 √ 32π log log N . Even more is known, in particular convergence of the full extremal process [BL16,BL18].…”

Section: The Membrane Modelmentioning

confidence: 99%

The maximum of the four-dimensional membrane model

Schweiger¹

2020

Ann. Probab.

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We show that the centred maximum of the four-dimensional membrane model on a box of sidelength N converges in distribution. To do so we use a criterion of Ding, Roy and Zeitouni and prove sharp estimates for the Green's function of the discrete Bilaplacian. These estimates are the main contribution of this work and might also be of independent interest. To derive them we use estimates for the approximation quality of finite difference schemes as well as results for the Green's function of the continuous Bilaplacian.

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Convergence in Law of the Maximum of the Two‐Dimensional Discrete Gaussian Free Field

Abstract: We consider the discrete two-dimensional Gaussian free field on a box of side length N , with Dirichlet boundary data, and prove the convergence of the law of the centered maximum of the field.

Cited by 132 publications

References 26 publications

1-stable fluctuations in branching Brownian motion at critical temperature I: The derivative martingale

1-stable fluctuations in branching Brownian motion at critical temperature I: The derivative martingale

Freezing and Decorated Poisson Point Processes

The maximum of the four-dimensional membrane model

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