We have recently reported the discovery of five low redshift Lyman continuum (LyC) emitters (LCEs, hereafter) with absolute escape fractions fesc(LyC) ranging from 6 to 13%, higher than previously found, and which more than doubles the number of low redshift LCEs.We use these observations to test theoretical predictions about a link between the characteristics of the Lyman-alpha (Lya) line from galaxies and the escape of ionising photons. We analyse the Lya spectra of eight LCEs of the local Universe observed with the Cosmic Origins Spectrograph onboard the Hubble Space Telescope (our five leakers and three galaxies from the litterature), and compare their strengths and shapes to the theoretical criteria and comparison samples of local galaxies: the Lyman Alpha Reference Survey, Lyman Break Analogs, Green Peas, and the high-redshift strong LyC leaker Ion2. Our LCEs are found to be strong Lya emitters, with high equivalent widths, EW(Lya)> 70 {AA}, and large Lya escape fractions, fesc(Lya) > 20%. The Lya profiles are all double-peaked with a small peak separation, in agreement with our theoretical expectations. They also have no underlying absorption at the Lya position. All these characteristics are very different from the Lya properties of typical star-forming galaxies of the local Universe. A subset of the comparison samples (2-3 Green Pea galaxies) share these extreme values, indicating that they could also be leaking. We also find a strong correlation between the star formation rate surface density and the escape fraction of ionising photons, indicating that the compactness of star-forming regions plays a role in shaping low column density paths in the interstellar medium of LCEs. The Lya properties of LCEs are peculiar: Lya can be used as a reliable tracer of LyC escape from galaxies, in complement to other indirect diagnostics proposed in the literature.
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