Context : Young brown dwarfs are known to possess circumstellar disks , a characteristic that is fundamental to the understanding of their formation process , and raises the possibility that these objects harbour planets .

Aims : We want to characterise the so far scarcely studied far-IR emission of disks around the young brown dwarf population of the \rho Ophiuchi cluster in LDN~1688 .

Methods : Recent observations of the \rho Ophiuchi cluster with the Herschel Space Observatory allow us to probe the spectral energy distribution ( SED ) of the brown dwarf population in the far-IR , where the disk emission peaks .
We performed aperture photometry at 70 , 100 , and 160 \mu m , and constructed SEDs for all previously known brown dwarfs detected .
These were complemented with ancillary photometry at shorter wavelengths .
We compared the observed SEDs to a grid of synthetic disks produced with the radiative transfer code MCFOST , and used the relative figure of merit estimated from the Bayesian inference of each disk parameter to analyse the structural properties .

Results : We detected 12 Class II brown dwarfs with Herschel , which corresponds to one-third of all currently known brown dwarf members of \rho Ophiuchi .
We do not detect any of the known Class III brown dwarfs .
Comparison to models reveals that the disks are best described by an inner radius between 0.01 and 0.07 AU , and a flared disk geometry with a flaring index between 1.05 and 1.2 .
Furthermore , we can exclude values of the disk scale-height lower than 10 AU ( measured at a fiducial radius of 100 AU ) .
We combined the Herschel data with recent ALMA observations of the brown dwarf GY92 204 ( ISO - Oph 102 ) , and by comparing its SED to the same grid of disk models , we derived an inner disk radius of 0.035 AU , a scale height of 15 AU with a flaring index of \beta \sim 1.15 , an exponent for dust settling of - 1.5 , and a disk mass of 0.001 M _ { \sun } .
This corresponds to a disk-to-central object mass ratio of \sim 1 \% .

Conclusions : The structural parameters constrained by the extended SED coverage ( inner radius and flaring index ) show a narrow distribution for the 11 young brown dwarfs detected in \rho Ophiuchi , suggesting that these objects share the same disk evolution and , perhaps , formation .