Introduction

The problem of bosons in the presence of disorder has generated much theoretical and experimental interest. The superfluid fraction in liquid $^4$He has been measured for different types of adsorbing porous media. In vycor, which has small ($\approx 70$ Å) pores and porosity of $30\%$ the superfluid transition is considerably suppressed, but exhibits the same critical exponent as in the bulk [7]. In contrast, in aerogel, which is characterized by larger pores with a broad distribution of sizes and porosity $85-99.5\%$, the superfluid transition is changed by only a few milli-kelvins while the critical exponents are quite different from the bulk [9,10]. Some experimental studies have also investigated, by measuring the dynamic structure factor, the nature of the elementary excitations in these systems [28] and the role played by the condensate fraction []. Theoretical studies of these effects have been proposed, mostly concerning models on a lattice. Many of the theoretical works address the problem of the superfluid-insulator transition and the critical behavior near the phase transition [30,8,31]. The boson localization and the structure of the Bose-glass phase have also been investigated. Quantum Monte Carlo techniques have been used for numerical simulations of the disordered Bose systems at low temperatures. Most of them concern systems on a lattice using the Bose-Hubbard or equivalent models. These studies have been carried out in 1D [32], 2D [21,33,34] and 3D [35,36] both at zero and finite temperatures. The structure of the phase diagram has been investigated and the properties of the superfluid, Mott insulator and Bose-glass phase have been addressed. There are very few simulations of disordered boson systems in the continuum. In ref. [37] the effect of impurities on the excitation spectrum in liqiud $^4$He is investigated using PIMC. The same technique is applied to study the effect of disorder on the superfluid transition in a Bose gas [38]. We apply DMC to study a hard-sphere Bose gas at zero temperature in the presence of hard-sphere quenched impurities. Hard sphere quenched impurities are easy are easy to implement in a numerical simulation and provide a reasonable model for liquid $^4$He in the porous media. Another possible physical realization of this model is given by trapped gases in the presence of heavy impurities. The free parameters in our simulations are:

• the density of the particles $na^3$, where $a$ is the diameter of the hard-sphere particle scattering length,
• the concentration of impurities $\chi = N^{imp}/N$ fixed by the ratio of the number of impurities to the number of particles used in the simulation,
• the ratio $b/a$, where $b = R^{imp}+a/2$ with $R^{imp}$ radius of the hard-sphere impurity.

The same parameters (2.8) were used in the perturbative analysis discussed previously (see section ). The goals of our study are:

• recover the ``weak'' disorder regime where the results of perturbation theory apply,
• verify the scaling behavior of the effects due to disorder in terms of the single parameter $R = \chi\,(b/a)^2$ as predicted from the Bogoliubov model
• understand if one can realize situations where the superfluid density is smaller than the condensate fraction
• investigate if within our model there exists a quantum phase transition for strong disorder