In case there will be zero space launches starting at some point in time. How long will it take for the space debris mass in LEO to be halved?
NASA has a decent FAQ section on their website about orbital debris.
How many orbital debris are currently in Earth orbit?
More than 21,000 orbital debris larger than 10 cm are known to exist. The estimated population of particles between 1 and 10 cm in diameter is approximately 500,000. The number of particles smaller than 1 cm exceeds 100 million.
How long will orbital debris remain in Earth orbit?
The higher the altitude, the longer the orbital debris will typically remain in Earth orbit. Debris left in orbits below 600 km normally fall back to Earth within several years. At altitudes of 800 km, the time for orbital decay is often measured in decades. Above 1,000 km, orbital debris will normally continue circling the Earth for a century or more.
Since most of the accounted for orbital debris is located in the 750-800 KM range, it would take several decades for orbital decay to significantly reduce the amount of man made space debris.
Anything below 300km will be out of orbit within a month or so, anything over 10,000km will be up there for essentially for ever. Dense objects will stay up for longer than low density objects (a sheet of tinfoil will slow faster than the same foil rolled up in a ball) and eccentric orbits decay more slowly than circular ones (because drag only happens at the low point of the orbit and does not greatly affect the height of the low point, it just tends to make the orbit more circular).
Pretty much most stuff under 600km will be clear in 40 years.
How this stacks up as a half life, well you could guess a distribution for the densities for the debris and work it out that way or you could just guess, i.e. mostly gone after five half lives (95% gone roughly) and 40/5 = a half life of 8 years for debris below 600km. But what's left will be your small dense really dangerous objects, i.e. bolts not flecks of paint. Plus there will be more stuff falling down from higher up and debris generated from collisions.
There's a piece of free software here: http://orbitaldebris.jsc.nasa.gov/mitigate/das.html that calculates orbital lifetimes if you are interested.
The rate of decay of space debris is driven primarily by drag from the Earth's atmosphere. The degree to which the atmosphere slows and object is based on the objects ballistic coefficient. This is because the kinetic energy of an object is the half objects mass and it's velocity^2, and the drag force is proportional to the wetted surface area, or the cross section area of the debris item. However this is only half the story.
The space debris problem is such a troublesome problem because objects can collide with one another. When these collisions happen objects are broken into smaller pieces. Generally this has a positive effect on the decay rate since the total wetted area of the pieces is greater than the wetted area of a intact object. However not all pieces are the same and hence you will get some very dense and some very low density objects. This will cause some object to have high ballistic coefficient and some to have low BCs, the pieces with the higher BCs will stay in orbit from longer than the other pieces and potentially longer than the original intact spacecraft. The net effect over time is an increase in the total number of objects, even if the total mass in orbit is decreasing.
Since this is a complex problem Monte Carlo simulations are often used to predict the rate of increase in debris flux (number of objects). However the uncertainties involved in these simulations leave a lot of unknown factors and different models predict different outcomes. Please add a comment if you'd like more specific information - a half life is not something that can accurately be predicted here.