We consider the discrete Gaussian Free Field in a square box in ( Z ^2 ) of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as ( N ). Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an r N -neighborhood thereof, we prove that the associated process tends, whenever ( r_N ) and ( r_N N 0 ), to a Poisson point process with intensity measure ( Z ( dx) e ^ - h d h ), where ( := 2 g ) with g: = 2 π and where Z(dx) is a random Borel measure on [0, 1]2. In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure Z is a version of the derivative martingale associated with the continuum Gaussian Free Field.
: In a recent posting @cite_25 , Arguin and Zindy have extended some of their conclusions to the DGFF. Notwithstanding, the question of pure atomicity for @math remains open.
@cite_25
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This paper is the continuation of a previous paper (H. Knupfer and C. B. Muratov, Comm. Pure Appl. Math. 66 (2013), 1129‐1162). We investigate the classical isoperimetric problem modified by an addition of a nonlocal repulsive term generated by a kernel given by an inverse power of the distance. In this work, we treat the case of a general space dimension. We obtain basic existence results for minimizers with sufficiently small masses. For certain ranges of the exponent in the kernel, we also obtain nonexistence results for sufficiently large masses, as well as a characterization of minimizers as balls for sufficiently small masses and low spatial dimensionality. The physically important special case of three space dimensions and Coulombic repulsion is included in all the results mentioned above. In particular, our work yields a negative answer to the question if stable atomic nuclei at arbitrarily high atomic numbers can exist in the framework of the classical liquid drop model of nuclear matter. In all cases the minimal energy scales linearly with mass for large masses, even if the infimum of energy cannot be attained. © 2014 Wiley Periodicals, Inc. We adopt the following convention: p, q, x, y are real numbers and n is a natural number. Next we state a number of propositions: (1) Let G, H be groups and h be a homomorphism from G to H. If h ·h−1 = idH and h−1 · h = idG, then h is an isomorphism. (2) For every subset X of I and for every point a of I such that X = ]a, 1] holds Xc = [0, a]. The paper was written during author’s post-doctoral fellowship granted by Shinshu University, Japan. This work has been partially supported by the CALCULEMUS grant HPRN-CT-200000102 and KBN grant 4 T11C 039 24. We describe a model independent search for physics beyond the standard model in lepton final states. We examine 117 final states using 1.1 fb @math of @math collisions data at @math TeV collected with the D0 detector. We conclude that all observed discrepancies between data and model can be attributed to uncertainties in the standard model background modeling, and hence we do not see any evidence for physics beyond the standard model. We study the partial transposition W Γ=(id⊗t)W∈M dn (ℂ) of a Wishart matrix W∈M dn (ℂ) of parameters (dn,dm). Our main result is that, with d→∞, the law of mW Γ is a free difference of free Poisson laws of parameters m(n±1) 2. Motivated by questions in quantum information theory, we also derive necessary and sufficient conditions for these measures to be supported on the positive half-line.