We study the two-point correlation function of a uniformly selected sample of 4,426 luminous optical quasars with redshift 2.9 \leq z \leq 5.4 selected over 4041 deg ^ { 2 } from the Fifth Data Release of the Sloan Digital Sky Survey .
We fit a power-law to the projected correlation function w _ { p } ( r _ { p } ) to marginalize over redshift space distortions and redshift errors .
For a real-space correlation function of the form \xi ( r ) = ( r / r _ { 0 } ) ^ { - \gamma } , the fitted parameters in comoving coordinates are r _ { 0 } = 15.2 \pm 2.7 h ^ { -1 } Mpc and \gamma = 2.0 \pm 0.3 , over a scale range 4 \leq r _ { p } \leq 150 h ^ { -1 } Mpc .
Thus high-redshift quasars are appreciably more strongly clustered than their z \approx 1.5 counterparts , which have a comoving clustering length r _ { 0 } \approx 6.5 h ^ { -1 } Mpc .
Dividing our sample into two redshift bins : 2.9 \leq z \leq 3.5 and z \geq 3.5 , and assuming a power-law index \gamma = 2.0 , we find a correlation length of r _ { 0 } = 16.9 \pm 1.7 h ^ { -1 } Mpc for the former , and r _ { 0 } = 24.3 \pm 2.4 h ^ { -1 } Mpc for the latter .
Strong clustering at high redshift indicates that quasars are found in very massive , and therefore highly biased , halos .
Following Martini & Weinberg , we relate the clustering strength and quasar number density to the quasar lifetimes and duty cycle .
Using the Sheth & Tormen halo mass function , the quasar lifetime is estimated to lie in the range 4 \sim 50 Myr for quasars with 2.9 \leq z \leq 3.5 ; and 30 \sim 600 Myr for quasars with z \geq 3.5 .
The corresponding duty cycles are 0.004 \sim 0.05 for the lower redshift bin and 0.03 \sim 0.6 for the higher redshift bin .
The minimum mass of halos in which these quasars reside is 2 - 3 \times 10 ^ { 12 } h ^ { -1 } M _ { \odot } for quasars with 2.9 \leq z \leq 3.5 and 4 - 6 \times 10 ^ { 12 } h ^ { -1 } M _ { \odot } for quasars with z \geq 3.5 ; the effective bias factor b _ { eff } increases with redshift , e.g. , b _ { eff } \sim 8 at z = 3.0 and b _ { eff } \sim 16 at z = 4.5 .