String theory tells us that stacks of D‑branes produce a space where the string coupling varies with the radial direction. For ordinary D‑branes this varying coupling is dual to the running of a field‑theory coupling. For D‑instantons — objects with no worldvolume space or time — the usual geometric picture is missing. This paper proposes two complementary matrix methods that recover the same running of the type IIB axio‑dilaton from the IKKT matrix model, a proposed non‑perturbative description of D‑instantons.

The first method uses a Coulomb branch construction. The authors split the full set of D‑instantons into two separated stacks with sizes N and M and distance Δ. In the matrix language this is done by giving diagonal matrices fixed values and integrating out the heavy off‑diagonal modes that represent strings stretching between the stacks. At leading order this produces a correction to the IKKT action of the form Tr[Y,Y]^4. IKKT had already computed the coefficient of this operator; for the D‑instanton case the coefficient is proportional to M/(Δ/α′)^8, where α′ is the string length squared.

The second method applies a matrix renormalization group (RG) idea due to Brézin and Zinn‑Justin (BZJ). That method reduces the matrix rank N step by step and, unlike the Coulomb branch procedure, also integrates over the relative position between blocks. Applying the BZJ flow to the IKKT partition function produces a running of the matrix coupling g with the rank N. The authors show that this N‑dependence matches the N‑dependence of the type IIB axio‑dilaton (the pair of fields combining the string coupling and a Ramond‑Ramond axion) seen in supergravity.

Why this matters is that it gives a concrete route toward “zero‑dimensional” holography: a map between a matrix model with no spacetime and a ten‑dimensional IIB background with a radial direction encoded in matrix data. The Coulomb branch calculation ties the coefficient of the leading correction in the matrix action to the harmonic function H that appears in the supergravity solution. The BZJ flow gives a complementary picture where the size N of the matrices itself plays the role of a renormalization scale and reproduces the expected dependence of the bulk dilaton and axion on N.