The paradigm of using multi-level and multi-link integration is a very crucial concept in doing Monte-Carlo calculations. Most of the simulations of Field Theories, or Stat-Mech systems, involve calculating expectation values of observables over the thermal ensemble determined by the action or the hamiltonian, as the case may be. Speaking loosely in the language of QFT, the expectation values of these observables are dominated by UV-contributions that appear at the level of the lattice scale ~1/a. These need to be integrated out carefully. One way is to collect as much statistics as you possibly can, and then average them over.
However, one can be clever, and try to integrate out only those fluctuations that occur at the scale ~1/a. This can be achieved, for example, in the case of the SU(3) lattice gauge theory, by integrating over the staples adjacent to a link, and keeping everything else in the problem (ie here in this case, links on other lattices) fixed. Thus, having integrated out the large scale fluctuations efficiently, one can do the averaging (integrals) over the rest of the lattice, giving a result that is much more stable. This is the so-called Multi-link method. Note that one can do the integration over the staples numerically (which means use another MC-integration step for this) or semi-analytically (which means that write the integral in quadrature in terms of some Special functions, and then evaluate this closed form integral numerically).
As often happens with lattice gauge theorists, you might want to take the continuum limit. This is the limit of large bare coupling β, ie small g2. In this limit, the gauge link matrices become close to unity and their fluctuations are rather small. So you'd expect the efficiency to decrease towards the continuum limit.
The Multi-level algorithm is a generalization of the former and can be used to overcome the aforementioned difficulty. In the Multi-level scheme, you divide the full lattice into several sub-lattices, say in the time direction. So if the thickness of the sub-lattice is d, then you have Nt/d sub-lattices in your disposal. The idea is the same before, just integrate over each of the sub-lattices, keeping the others fixed. Of course, in this case, no semi-analytical formula can be written down, and this procedure has to be performed numerically. In this case, note that you will be integrating over contributions in the regime ~1/(d*a); and this has often the effect of better averaging of the observables, ie less noise. Note that the hallmark of a calculation with the Multi-level scheme is to decrease errors exponentially. In particular, this means that even if you are measuring a correlation function that decays exponentially, one can maintain a fixed error/mean ratio for the entire correlation function!
The crucial requirement, which ensures that the multi-level scheme will work for the gauge theory, is the locality of the action. With fermions present, this is not possible in the standard scheme of doing things, where the fermions are integrated over and written as a determinant in the partition function. Since the determinant is a non-local object, spanning the whole of the lattice, the division into different length scales cannot be made, and multi-level scheme cannot be applied.
However, one can be clever, and try to integrate out only those fluctuations that occur at the scale ~1/a. This can be achieved, for example, in the case of the SU(3) lattice gauge theory, by integrating over the staples adjacent to a link, and keeping everything else in the problem (ie here in this case, links on other lattices) fixed. Thus, having integrated out the large scale fluctuations efficiently, one can do the averaging (integrals) over the rest of the lattice, giving a result that is much more stable. This is the so-called Multi-link method. Note that one can do the integration over the staples numerically (which means use another MC-integration step for this) or semi-analytically (which means that write the integral in quadrature in terms of some Special functions, and then evaluate this closed form integral numerically).
As often happens with lattice gauge theorists, you might want to take the continuum limit. This is the limit of large bare coupling β, ie small g2. In this limit, the gauge link matrices become close to unity and their fluctuations are rather small. So you'd expect the efficiency to decrease towards the continuum limit.
The Multi-level algorithm is a generalization of the former and can be used to overcome the aforementioned difficulty. In the Multi-level scheme, you divide the full lattice into several sub-lattices, say in the time direction. So if the thickness of the sub-lattice is d, then you have Nt/d sub-lattices in your disposal. The idea is the same before, just integrate over each of the sub-lattices, keeping the others fixed. Of course, in this case, no semi-analytical formula can be written down, and this procedure has to be performed numerically. In this case, note that you will be integrating over contributions in the regime ~1/(d*a); and this has often the effect of better averaging of the observables, ie less noise. Note that the hallmark of a calculation with the Multi-level scheme is to decrease errors exponentially. In particular, this means that even if you are measuring a correlation function that decays exponentially, one can maintain a fixed error/mean ratio for the entire correlation function!
The crucial requirement, which ensures that the multi-level scheme will work for the gauge theory, is the locality of the action. With fermions present, this is not possible in the standard scheme of doing things, where the fermions are integrated over and written as a determinant in the partition function. Since the determinant is a non-local object, spanning the whole of the lattice, the division into different length scales cannot be made, and multi-level scheme cannot be applied.
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