In "A Rose for Emily" by William Faulkner, what is the role of dust and decay in the story?

Symbolic of the demise of the Old South and the way of life that Emily has known, dust and decay add to the gothic effect of Faulkner's story, as well.  In "A Rose for Emily," after the men of Colonel Sartoris's generation have passed away, a new delegation of aldermen with "more modern ideas" call upon Emily to request her taxes:

A deputation...knocked at the door through which no visitor had passed since she ceased giving china-painting lessons eight or ten years earlier.  They were admitted by the old Negro into a dim hall....It smelled of dust and disuse--a close, dank smell.  When...they sat down, a faint dust rose sluggishly about their things, spinning with slow motes in the single sunray.

Amid this dust Emily enters dressed in black, wearing her deceased father's watch chain.  In the midst of the dying world, Emily clings to what she has known and asks the old servant to "Show these gentlemen out."  Emily insists that her father is not dead, and "clings to that which had robbed her, as people will."

Likewise, Emily clings to another who has "robbed" her.  Having lost the respect of the townspeople by going with a worker from the North, Homer Barron, Emily remains reclusive until she dies,

Fell ill in the house filled with dust and shadows...

After Emily is buried, the narrators force their way into the one room that no one has seen in forty years:

The violence of breaking down the door seemed to fill this room with prervading dust.  A thin, acrid pall as of the tomb seemed to lie everywhere upon this room decked and furnished as for a bridal....Thne we noticed that in the second pillow was the indentation of a head. One of us lifeted someting from it...that faint and invisible dust dry and acrid in the nostrils....

The remnant of Emily in her strand of hair, the remant of her life in the dry dust, a life thwarted by time and patriarchy--a truly Southern Gothic moment.  Like the Old South and its way of life, Emily has also decayed, leaving only a strand of hair as a reminder of what once was.