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Non-uniform black strings (NUBS) are studied by the large $D$ effectivetheory approach. By solving the near-horizon geometry in the $1/D$ expansion,we obtain the effective equation for the deformed horizon up to thenext-to-next-to-leading order (NNLO) in $1/D$. We also solve the far-zonegeometry by the Newtonian approximation. Matching the near and far zones, thethermodynamic variables are computed in the $1/D$ expansion. As the result, thelarge $D$ analysis gives a critical dimension $D_*\simeq13.5$ at which thetranslation-symmetry-breaking phase transition changes between first and secondorder. This value of $D_*$ agrees perfectly, within the precision of the $1/D$expansion, with the result previously obtained by E. Sorkin through thenumerical resolution. We also compare our NNLO results for the thermodynamicsof NUBS to earlier numerical calculations, and find good agreement within theexpected precision.

Non-uniform black strings and the critical dimension in the $1/D$ expansion