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(Abridged) Long Gamma-Ray Bursts (GRBs) offer a promising tool to trace thecosmic history of star formation, especially at high redshift whereconventional methods are known to suffer from intrinsic biases. Previousstudies of GRB host galaxies at low redshift showed that high surface densitiesof stellar mass and star formation rate (SFR) can potentially enhance the GRBproduction. We assess how the size, the stellar mass and SFR surface densitiesof distant galaxies affect their probability to host a long GRB, using a sampleof GRB hosts at $z > 1$ and a control sample of star-forming sources from thefield. We gather a sample of 45 GRB host galaxies at $1 < z < 3.1$ observedwith the Hubble Space Telescope WFC3 camera in the near-infrared. Using theGALFIT parametric approach, we model the GRB host light profile and derive thehalf-light radius for 35 GRB hosts, which we use to estimate the SFR andstellar mass surface densities of each object. We compare the distribution ofthese physical quantities to the SFR-weighted properties of a complete sampleof star-forming galaxies from the 3D-HST deep survey at comparable redshift andstellar mass. We show that, similarly to $z < 1$, GRB hosts are smaller in sizeand they have higher stellar mass and SFR surface densities than field galaxiesat $1 < z < 2$. Interestingly, this result is robust even when consideringseparately the hosts of GRBs with optically-bright afterglows and the hosts ofdark GRBs. At $z > 2$ though, GRB hosts appear to have sizes and stellar masssurface densities more consistent with those characterizing the field galaxies.In addition to a possible trend toward low metallicity environment, otherenvironmental properties such as stellar density appears to play a role in theformation of long GRBs, at least up to $z \sim 2$. This might suggest that GRBsrequire special environments to be produced.

Are the host galaxies of Long Gamma-Ray Bursts more compact than star-forming galaxies of the field?