Consider nearest-neighbor oriented percolation in \(d+1\) space–time dimensions. Let \(\rho ,\eta ,\nu \) 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 \(d\nu \ge \eta +2\rho \), which holds for all \(d\ge 1\) and is a strict inequality above the upper-critical dimension 4, becomes an equality for \(d=1\), i.e., \(\nu =\eta +2\rho \), provided existence of at least two among \(\rho ,\eta ,\nu \). The key to the proof is the recent result on the critical box-crossing property by Duminil-Copin et al. .
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This work was initiated when I started preparation for the Summer School in Mathematical Physics, held at the University of Tokyo from August 25 through 27, 2017. I am grateful to the organizers, Yasuyuki Kawahigashi and Yoshiko Ogata, for the opportunity to speak at the summer school and meet with many researchers in the laminar-turbulent flow transition. Finally, I would like to thank Alessandro Giuliani for his support during the refereeing process and anonymous referees for valuable comments to the earlier version to this paper.
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