Jung wrote a lot about how evolutionary pressures shaped the development of the human understandings of the life cycle, and the archetypes he described are present in almost every myth to the extent that they seem familiar regardless of minor cultural variations. I'll have to over-simplify to describe them, but overall, each archetype can be identified most easily through generalities, with specifics not quite capturing the full expression of a single archetype, especially given that a man and a woman experience each other as entities in different ways (called the anima and animus for each sex respectively).

The symbolism behind the Father figure is pretty well known; he is the archetype of the kingly figure who dies for his children and people as personified in the Adonis myth and the idea of sacrificing the king to save the land. There are often conflicting roles between the Father, the King and the Priest because the leaders/strongmen/strongwomen of most early societies fulfilled dual roles as societal leaders, war chiefs and high priests. Fu Hao, for example, did all of those things as a Battle Queen in Bronze Age China, and in regards to male Priest Kings we often deal with the same idea in Welsh, Arthurian and even Christian myths (especially that of King David).

The Wanderer is pretty similar to the female Virgin or the Huntress; this archetype has no set dwelling place and because of this has the most freedom. In most myths, he can be compared to a follower of Apollo, where he is a self-centered individual (not necessarily in a bad way), but in other myths, he is the trickster, warrior or a rogue. In Greek myths, the Prince Paris was said to be a shepherd that was adopted by the King of Troy, so he could be considered to fit the mold of this archetype. Because he is a bachelor, this is the most romantic of the three types. This is the survivor that finds themselves on adventures on the behalf of the King/Queen in a bid to prove his worth.

The Wise Old Man is the precursor to the Wizard. Pretty similar to the Crone - since he is past his most potent years sexually and physically, his means of dealing with the world revolves around information and secret knowledge. Since he's near the end of his life, he's free of the egotistical self-consciousness that limits and prevents the young from reaching anything like true enlightenment. In most myths, this personification is usually the most mature due to their experience or personality and is usually removed from the hero in order to allow the male hero to come into his own. This is very similar to how a heroine gains secret knowledge from the Wise Old Woman/Crone that she uses to go to another phase of her life.

Those descriptions are pretty rough, but the archetypes are represented in the vast majority of cultural myths to the extent that cultural variations tend to support rather than conflict with the overall descriptions C.G. Jung applied to them.

I would say Rand, being connected to the creative force through his sexual potency (often recognized as being magical in the older myths) is the Father, Mat is the Wanderer because freedom, fear of women (i.e. fear of progressing to a creative phase of his life) and the ability to avoid being tied down is one of his major character traits while Perrin is the closest to being a Wise Old Man because his struggle is more about applying technique to create tools versus embracing his innate nature as a sexual being himself.