Subleading Corrections, Double Scaling, and Nonuniform Limits

A fixed large-$N$ expansion is controlled only when its coefficients remain uniform in every other parameter being varied. Near a matrix support transition, an infrared singularity, or a competing saddle, nominally subleading terms can become leading. Double scaling deliberately approaches such a boundary with $N$ so that infinitely many $1/N$ orders survive; effects of order $e^{-cN}$ remain invisible to every fixed order and require separate data.

Required background. Large-N limits, normalizations, and orders of limits supplies the held-coupling and limit-order prescription. Matrix eigenvalue saddles and phase transitions supplies the normalized quartic density and exact support boundary used below.

Helpful background. Multi-saddle sums and dilute ensembles supplies the distinction between fluctuations about one saddle and additional exponential sectors.

Fixed-order error estimates

Let a normalized observable at fixed regulator, volume, state, and couplings have

$$\mathcal O_N(\delta) \sim \sum_{k=0}^{K-1} \frac{\mathcal O_k(\delta)}{N^{pk}} +R_K(N,\delta).$$

The step size is model-dependent: $p=2$ for the closed orientable genus expansion, often $p=1$ for vector singlets, and a tensor model is graded by its own degree. The first correction is small only if

$$\epsilon_1(N,\delta) = \frac1{N^p} \left\lvert \frac{\mathcal O_1(\delta)}{\mathcal O_0(\delta)} \right\rvert \ll1.$$

If $\mathcal O_1/\mathcal O_0\sim\delta^{-q}$ near a boundary $\delta=0$, fixed-order control requires

$$\delta\gg N^{-p/q}.$$

The joint region $\delta\sim N^{-p/q}$ is nonuniform: the correction is $O(1)$ even though $N$ is large. Quoting “suppressed by $1/N^p$” without bounding its coefficient is incomplete.

For a matrix free energy away from criticality,

$$\log Z_N \sim \sum_{h=0}^{\infty} N^{2-2h}F_h(\delta).$$

This formula orders each fixed genus. It neither proves convergence of the sum over $h$ nor controls $F_h$ as $\delta\to0$.

First application: the quartic support-merging window

Use the normalized quartic matrix model

$$V(x) = \frac r2x^2+\frac g4x^4, \qquad g>0,$$

whose large-$N$ support changes at

$$r_c=-2\sqrt g.$$

Define the dimensionless distance

$$\tau = \frac{r-r_c}{\sqrt g} = \frac{r+2\sqrt g}{\sqrt g}.$$

On the one-cut side, the density is

$$\rho_1(x) = \frac1{2\pi} \left(gx^2+c\right) \sqrt{a^2-x^2}, \qquad c=r+\frac{ga^2}{2}.$$

Differentiating $4ra^2+3ga^4=16$ at the critical point gives

$$\left.\frac{\mathrm d a^2}{\mathrm dr}\right|_{r_c} = -\frac1g, \qquad c=\frac12(r-r_c)+O\!\left((r-r_c)^2\right).$$

Exactly at criticality,

$$\rho_c(x) \sim \frac{a_cg}{2\pi}x^2 \qquad(x\to0).$$

The expected number of eigenvalues in $[-x,x]$ is therefore of order $Nx^3$. The microscopic scale at the merging point is

$$x_{\mathrm{micro}}\sim N^{-1/3}.$$

On the two-cut side, the inner endpoint satisfies

$$\alpha^2 = \frac{-r-2\sqrt g}{g} = -\frac{\tau}{\sqrt g}.$$

The gap becomes comparable to the microscopic scale when $\alpha\sim N^{-1/3}$, or

$$\tau\sim N^{-2/3}.$$

Thus the support-merging double-scaling variable is

$$\kappa = N^{2/3}\tau = N^{2/3} \frac{r+2\sqrt g}{\sqrt g},$$

held fixed as $N\to\infty$ and $r\to r_c$. Fixed nonzero $\tau$ gives an ordinary one-cut or two-cut expansion; fixed $\kappa$ resolves the rounded critical window and resums infinitely many nonuniform orders. The associated quartic critical asymptotics are governed by a Painlevé II scaling problem Bleher and Its 2005, §§1–2.

Shared phase map. The eigenvalue support and phase map fixes $g=1$ and displays the one-cut, critical, and two-cut densities whose local scales were compared here.

Shared comparison. The large-N scaling comparison states why this joint critical limit is distinct from the fixed-phase $N\to\infty$ limit.

Double scaling is a new limit, not a correction term

Suppose the singular genus coefficients behave as

$$F_h^{\mathrm{sing}}(\tau) \sim \tau^{\gamma(1-h)}.$$

If $\tau=N^{-2/\gamma}\kappa$, then

$$N^{2-2h}F_h^{\mathrm{sing}}(\tau) \sim \kappa^{\gamma(1-h)}$$

at every $h$. No finite truncation in genus remains ordered. One instead seeks a scaling function with its own differential equation, boundary conditions, and asymptotic sectors.

In the quartic merging problem, $\tau\sim N^{-2/3}$ corresponds to $\gamma=3$. Other critical points have other susceptibility exponents and therefore other scaling variables. The pure-gravity one-matrix critical point, for example, has its own double-scaling exponent and Painlevé I equation. One cannot infer the exponent solely from the phrase “matrix criticality.”

The original double-scaling constructions show how a continuum sum over genera emerges from a correlated matrix-size and coupling limit Brézin and Kazakov 1990, pp. 144–150 and Douglas and Shenker 1990, §§2–4, pp. 640–654.

Infrared enhancement and limit order

Nonuniformity is not confined to matrix models. A nominal vector correction can contain

$$\frac{\mu^{4-d}}N \int\frac{\mathrm d^dp}{(2\pi)^d} \frac1{(p^2+m^2)^2}.$$

Here the fixed reference scale $\mu$ makes the displayed correction dimensionless. For $d<4$, it grows as $N^{-1}(\mu/m)^{4-d}$ when $m\to0$. The large-$N$ estimate is uniform only if

$$N\left(\frac m\mu\right)^{4-d}\gg1.$$

Taking $N\to\infty$ at fixed $m>0$ and then $m\to0$ need not equal a joint critical limit with $N(m/\mu)^{4-d}$ fixed. The same issue arises with growing volume, nearly massless collective modes, and saddle Hessian eigenvalues approaching zero.

At a practical level, every subleading claim should identify:

• the norm or observable in which the remainder is measured;
• the parameter domain on which its coefficient is bounded;
• the regulator, state, and volume held fixed;
• the distance to critical or symmetry-breaking boundaries;
• whether the estimate is pointwise or uniform.

Exponentially small sectors and large order

An expansion in $1/N$ cannot see

$$\Delta\mathcal O_N \sim e^{-NA(\delta)} \sum_{k\ge0}\frac{c_k(\delta)}{N^k}.$$

At fixed $A>0$, this is smaller than every power. If $A(\delta)\to0$ near a saddle collision, it can become comparable to perturbative terms in a joint limit. Such sectors can also control the large-order growth of $F_h$ even when they are numerically tiny at low genus.

Therefore:

• asymptotic accuracy at a fixed truncation does not imply convergence;
• a zero result at every fixed order does not prove an exact zero;
• Borel ambiguity or factorial growth signals missing completion data, not a license to choose a sum arbitrarily;
• double scaling may itself require transseries boundary conditions.

Large-$N$ instanton actions and their relation to large-order behavior are analyzed in Mariño 2015, ch. 10, pp. 301–325. Chapter 10 of this volume develops the transseries completion; this page owns only the diagnosis that fixed $1/N$ reasoning has failed.

Common pitfalls

Estimating only the explicit power of $N$. A divergent coefficient can cancel the suppression. Bound the full ratio in the parameter region used.

Calling double scaling “including the next correction.” It keeps infinitely many orders and defines a new correlated limit.

Concluding exactness from all perturbative coefficients. Contributions proportional to $e^{-cN}$ vanish to every algebraic order.

Exercises

1. If the first relative correction is $N^{-2}\delta^{-5}$, find the fixed-order control region and double-scaling variable.

Solution

Control requires $N^{-2}\delta^{-5}\ll1$, or $\delta\gg N^{-2/5}$. A correlated boundary variable is $\kappa=N^{2/5}\delta$.

2. Derive the $N^{-1/3}$ eigenvalue scale from $\rho_c(x)\propto x^2$.

Solution

The expected number in $[-x,x]$ scales as

$$N\int_{-x}^{x}\mathrm dy\,\rho_c(y) \propto Nx^3.$$

Setting this number to $O(1)$ gives $x\sim N^{-1/3}$.

3. For $d=3$, when is $\mu N^{-1}\int\mathrm d^3p\,(p^2+m^2)^{-2}$ small?

Solution

The integral scales as $m^{-1}$, so the dimensionless correction is of order $\mu/(Nm)$. It is small when $Nm/\mu\gg1$; a joint limit with $Nm/\mu$ fixed is nonuniform.

References

• Bleher, P. M., and Its, A. R. (2005). “Asymptotics of the Partition Function of a Random Matrix Model.” Annales de l’Institut Fourier 55, 1943–2000. doi:10.5802/aif.2147. Open PDF.
• Brézin, E., and Kazakov, V. A. (1990). “Exactly Solvable Field Theories of Closed Strings.” Physics Letters B 236, 144–150. doi:10.1016/0370-2693(90)90818-Q.
• Douglas, M. R., and Shenker, S. H. (1990). “Strings in Less Than One Dimension.” Nuclear Physics B 335, 635–654. doi:10.1016/0550-3213(90)90522-F.
• Mariño, M. (2015). Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press. doi:10.1017/CBO9781107705968.