Three- and two-point spatial correlations of intergalactic medium at $z\sim 2$ using projected quasar triplets

We present analysis of two- and three-point correlation functions of Ly$\alpha$ forest (at $2\le z\le 2.5$) using X-Shooter spectra of three background quasar triplets probing transverse separations of 0.5-1.6 pMpc. We present statistics based on transmitted flux and clouds identified using Voigt profile fitting. We show that the observed two-, three-point correlation functions and reduced three-point correlation (i.e Q) are well reproduced by our simulations. We assign probabilities for realising all the observed correlation properties simultaneously using our simulations. Our simulations suggest an increase in correlation amplitudes and Q with increasing $N_{\rm HI}$. We roughly see this trend in the observations too. We identify a concurrent gap of 17$\mathring{A}$ (i.e 14.2 $h^{-1}$cMpc, one of the longest reported) wide in one of the triplets. Such gap is realised only in 14.2% of our simulated sightlines and most of the time belongs to a void in the matter distribution. In the second triplet, we detect DLAs along all three sightlines (with spatial separations 0.64 to 1.6 pMpc) within a narrow redshift interval (i.e $\Delta z = 0.088$). Detection of a foreground quasar ($\sim$ 1 pMpc from the triplet sightlines) and excess partial Lyman Limit systems around these DLAs suggest that we may be probing a large over-dense region. We also report positive CIV- CIV correlations up to $\sim 500$ $km s^{-1}$ only in the longitudinal direction. Additionally, we conclude a positive CIV-Ly$\alpha$ correlations for higher $N_{\rm HI}$ thresholds up to a scale of $\sim 1000$ $km s^{-1}$ both in transverse and longitudinal directions.