Finite-N Smoothing, Tunneling, and Metastability

At infinite N, the minimum of two $O(N^2)$ saddle free energies can have a sharp kink. At finite N on a compact spatial manifold, the exact thermal trace is normally analytic: both saddle sectors mix, the crossover has finite width, metastable phases decay, and the discrete spectrum ultimately produces recurrences. The relevant corrections are not all the same—rounding is algebraic near the crossing, while tunneling can be exponentially suppressed.

Required background. Approximate Bulk Locality from Spectral and Mellin Data fixes the finite-N claim ceiling. Euclidean Saddles, Thermal States, and Hawking–Page Transitions supplies the competing large-N saddles.

Helpful background. Black-Hole Instabilities and New Phases separates linear instability from decay. Metastability, Spinodals, and Thermal Decay supplies nucleation theory. Corrections, Nonuniform Limits, and Failure Modes fixes the order-of-limits discipline.

First application. Estimate the N scaling of tunneling between thermal AdS and black-hole saddles and contrast it with the infinite-N free-energy crossing.

Two-saddle rounding near a crossing

Suppose the leading finite-N approximation is

$$Z(T)\simeq e^{-N^2 f_1(T)}+e^{-N^2 f_2(T)}.$$

The exact effective free energy per $N^2$ is

$$f_N(T)=-\frac{1}{N^2}\log Z =\min(f_1,f_2) -\frac{1}{N^2}\log\!\left(1+e^{-N^2\lvert f_1-f_2\rvert}\right).$$

If $f_1-f_2\simeq-\Delta s\,(T-T_c)$, both saddles contribute when

$$\lvert T-T_c\rvert \lesssim \frac{1}{N^2\lvert\Delta s\rvert}.$$

Thus the rounding window in this two-saddle model is $O(N^{-2})$. Away from it, the subdominant equilibrium weight is exponentially small. This distinction prevents the common mistake of calling every finite-N effect $e^{-N^2}$.

Tunneling and metastable decay

A locally stable but globally subdominant phase can decay by a critical fluctuation or Euclidean bounce. When the gravitational action scales as $N^2$, the rate has the schematic form

$$\Gamma\sim A\,e^{-N^2 B},$$

where $B$ is an action difference and $A$ contains determinants, zero modes, and powers of N. The exponent must be computed in the declared ensemble and geometry. A Hawking–Page free-energy difference is not automatically the bounce action.

At a spinodal, a local minimum disappears and the barrier vanishes; near coexistence, a barrier may remain even when the two phases have equal free energy. Linear instability, nucleation, and equilibrium mixing therefore govern different regions and time scales.

Compact volume, infinite volume, and limit order

For a CFT on compact $S^{d-1}$ with a discrete spectrum and suitable convergence,

$$Z(\beta)=\sum_n e^{-\beta E_n}$$

is analytic for $\operatorname{Re}\beta>0$. A true nonanalyticity emerges only in a limit such as $N\to\infty$. On noncompact boundary space, an independent thermodynamic-volume limit can support phase transitions even at finite N. “Finite N smooths the transition” is therefore a compact-volume statement unless the volume limit is specified.

The order of N and time limits also matters. A single black-hole saddle predicts dissipative quasinormal decay. A finite-N discrete spectrum produces late-time fluctuations and recurrences. Taking $N\to\infty$ before $t\to\infty$ removes effects that can be exponentially late in the entropy.

Adversarial controls

Use only the dominant saddle. Replacing the sum by $e^{-N^2\min f_i}$ manufactures a kink at every N. Keeping both terms demonstrates the rounding directly.

Equate equilibrium weight and transition rate. The factor suppressing a subdominant saddle in $Z$ is determined by its free energy; the decay rate is determined by a transition configuration and fluctuation determinant. They need not share an exponent.

Ignore the spatial-volume limit. Planar systems can remain nonanalytic at finite N after infinite volume. A sphere and a plane do not have the same smoothing statement.

What remains controlled

The two-saddle sum determines the leading crossover width and equilibrium weights when no omitted saddle is comparable. A computed bounce can determine a metastable decay exponent in its semiclassical regime. Neither establishes the exact finite-N spectrum, the complete set of tunneling channels, or the very-late-time behavior of a fixed CFT.

The relation between the AdS thermal transition and gauge-theory confinement is the semiclassical result of Witten 1998; long-time recurrences and finite-entropy limits require the distinct analysis emphasized by Barbon and Rabinovici 2003.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Barbon, José L. F., and Eliezer Rabinovici. “Very Long Time Scales and Black Hole Thermal Equilibrium.” Journal of High Energy Physics 2003, 047 (2003). DOI; arXiv:hep-th/0308063.
• Maldacena, Juan. “Eternal Black Holes in Anti-de Sitter.” Journal of High Energy Physics 2003, 021 (2003). DOI; arXiv:hep-th/0106112.
• Witten, Edward. “Anti-de Sitter Space, Thermal Phase Transition, and Confinement in Gauge Theories.” Advances in Theoretical and Mathematical Physics 2, 505–532 (1998). DOI; arXiv:hep-th/9803131.