  We observe that the large $N$ world sheet RG in $c=1$ matrix model,
formulated in hep-th/0310106, hep-th/0311177, with $N^2$ quantum mechanical
degrees of freedom at small compactification radius is capable of capturing
dimensional mutation. This manifests in deforming the familiar $AdS_2$ quantum
mechanics in the minisuperspace Wheeler-de Witt (WdW) cosmology of the 2D
quantum gravity, obtained by the large $N$ RG with $N$ quantum mechanical
degrees of freedom only, to a modified WdW cosmology describing tunneling to an
inflationary de Sitter vacuum and its evolution. The scale fluctuation plays an
important role in providing an ansatz for uniquely choosing the initial wave
function. We observe that the nonperturbative effects due to the $N^2$ quantum
mechanical degrees of freedom introduce explicit open string moduli dependence
in the wave function via the Hubble scale, which determines the geometry of the
true vacuum one tunnels to. The modified WdW equation also captures controlled
formation of baby universes of vanishing size that self-tunes the de Sitter
cosmological constant to be small positive.
