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Register a new userHyperscaling for oriented percolation in 1+1 space-time dimensions
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Akira Sakai
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Consider nearest-neighbor oriented percolation in $d+1$ space-time dimensions. Let $rho,eta, u$ be the critical exponents for the survival probability up to time $t$, the expected number of vertices at time $t$ connected from the space-time origin, and the gyration radius of those vertices, respectively. We prove that the hyperscaling inequality $d ugeeta+2rho$, which holds for all $dge1$ and is a strict inequality above the upper-critical dimension 4, becomes an equality for $d=1$, i.e., $ u=eta+2rho$, provided existence of at least two among $rho,eta, u$. The key to the proof is the recent result on the critical box-crossing property by Duminil-Copin, Tassion and Teixeira (2017).
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We provide a complete proof of the diagrammatic bounds on the lace-expansion coefficients for oriented percolation, which are used in [arXiv:math/0703455] to investigate critical behavior for long-range oriented percolation above 2min{alpha,2} spatial dimensions.
Recently, Holmes and Perkins identified conditions which ensure that for a class of critical lattice models the scaling limit of the range is the range of super-Brownian motion. One of their conditions is an estimate on a spatial moment of order higher than four, which they verified for the sixth moment for spread-out lattice trees in dimensions $d>8$. Chen and Sakai have proved the required moment estimate for spread-out critical oriented percolation in dimensions $d+1>4+1$. We prove estimates on all moments for the spread-out critical contact process in dimensions $d>4$, which in particular fulfills the spatial moment condition of Holmes and Perkins. Our method of proof is relatively simple, and, as we show, it applies also to oriented percolation and lattice trees. Via the convergence results of Holmes and Perkins, the upper bounds on the spatial moments can in fact be promoted to asymptotic formulas with explicit constants.
We construct Holographic Space-time models that reproduce the dynamics of $1 + 1$ dimensional string theory. The necessity for a dilaton field in the $1 + 1$ effective Lagrangian for classical geometry, the appearance of fermions, and even the form of the universal potential in the canonical $1$ matrix model, follow from general HST considerations. We note that t Hoofts ansatz for the leading contribution to the black hole S-matrix, accounts for the entire S-matrix in these models in the limit that the string scale coincides with the Planck scale, up to transformations between near horizon and asymptotic coordinates. These $1 + 1$ dimensional models are describable as decoupling limits of the near horizon geometry of higher dimensional extremal black holes or black branes, and this suggests that deformations of the simplest model are equally physical. After proposing a notion of relevant deformations, we describe deformations, which contain excitations corresponding to linear dilaton black holes, some of which can be considered as UV completions of the CGHS model. We study the question of whether the AMPS paradox can be formulated in those models. It cannot, because the classical in-fall time to the singularity of linear dilaton black holes, is independent of the black hole mass. This result is reproduced by our HST models. We argue that it is related to the absence of quasi-normal modes of these black hole solutions, which is itself related to the fact that the horizon has zero area. This is compatible with the resolution of the AMPS paradox proposed in previous work with Fischler, according to which the compatibility conditions of HST identify the long non-singular sojourn of observers behind the horizon, with the dynamics of equilibration on the horizon as seen by a detector which has not yet fallen through the horizon.
In this paper we study stationary last passage percolation (LPP) in half-space geometry. We determine the limiting distribution of the last passage time in a critical window close to the origin. The result is a new two-parameter family of distributions: one parameter for the strength of the diagonal bounding the half-space (strength of the source at the origin in the equivalent TASEP language) and the other for the distance of the point of observation from the origin. It should be compared with the one-parameter family giving the Baik--Rains distributions for full-space geometry. We finally show that far enough away from the characteristic line, our distributions indeed converge to the Baik--Rains family. We derive our results using a related integrable model having Pfaffian structure together with careful analytic continuation and steepest descent analysis.
We prove that the Fourier transform of the properly-scaled normalized two-point function for sufficiently spread-out long-range oriented percolation with index alpha>0 converges to e^{-C|k|^{alphawedge2}} for some Cin(0,infty) above the upper-critical dimension 2(alphawedge2). This answers the open question remained in the previous paper [arXiv:math/0703455]. Moreover, we show that the constant C exhibits crossover at alpha=2, which is a result of interactions among occupied paths. The proof is based on a new method of estimating fractional moments for the spatial variable of the lace-expansion coefficients.
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