Conditional Gaussian Multiplicative Chaos

• Conditional Gaussian Multiplicative Chaos is a framework that defines random measures in critical regimes where standard GMC diverges by employing stochastic shifts with a random base reference.
• It captures the scaling limits of partition functions in critical polymer models through a cascade structure that ensures compatibility with subcritical GMC shifts.
• The approach has significant implications for modeling (2+1)-dimensional continuum polymers and stochastic heat equations, where log-correlated fields induce singular behavior.

Conditional Gaussian multiplicative chaos (conditional GMC) is a probabilistic structure underlying certain random measures associated with critical weak-disorder limits in models such as the continuum directed polymer on the diamond fractal of Hausdorff dimension two. In this context, standard (subcritical) GMC constructions fail due to divergence issues at criticality, but a conditional GMC structure persists, linking families of random measures through stochastic “shifts” with random base references. This framework characterizes the scaling limits of partition functions in critical polymer models and is conjectured to extend to (2+1)-dimensional critical continuum polymers and stochastic heat equations, where log-correlated fields are prevalent (Clark, 2019).

1. Gaussian Multiplicative Chaos in Classical and Critical Regimes

In the foundational setup, consider a measurable space $(\Gamma, \mu)$ (e.g., a fractal of dimension $d$ and its uniform measure) and a centered Gaussian field $\mathbf W = \{\mathbf W(p): p \in \Gamma\}$ with covariance $T(p, q) = \mathbb E[\mathbf W(p)\mathbf W(q)]$. The aim is to construct random measures via

$$\mathbf M_\beta(dp) = \exp\left(\beta \mathbf W(p) - \tfrac{\beta^2}{2} T(p, p)\right) \mu(dp),$$

for inverse temperature parameter $\beta \ge 0$. Since the field is typically distribution-valued, pointwise exponentiation is not well-defined. The classical Kahane theory, advanced by Shamov and others, provides meaning for $\mathbf M_\beta$ via regularization and limit procedures, especially in the subcritical regime ($\beta < \beta_c$), where $\beta_c$ is a model-dependent critical value.

In the subcritical regime, one can regularize $\mathbf W$ (via mollification, projection, or truncation) and obtain nontrivial measures with all positive moments finite, satisfying

$d$0

However, in the critical regime ($d$1), the procedure breaks down: the limit of $d$2 as $d$3 is almost surely zero, and the second moment $d$4 diverges.

2. Failure of Subcritical GMC at Critical Dimension on the Diamond Fractal

In the diamond hierarchical lattice (DHL) setting, the Hausdorff dimension is $d$5 for branching number $d$6 and segmentation parameter $d$7. For $d$8, $d$9 (subcritical). Here, white-noise coupling produces a true subcritical GMC.

Aggravatingly, at $\mathbf W = \{\mathbf W(p): p \in \Gamma\}$0 ($\mathbf W = \{\mathbf W(p): p \in \Gamma\}$1), log-divergences in the path intersection kernel $\mathbf W = \{\mathbf W(p): p \in \Gamma\}$2 cause the second moment to diverge for all $\mathbf W = \{\mathbf W(p): p \in \Gamma\}$3: $\mathbf W = \{\mathbf W(p): p \in \Gamma\}$4 This implies that for any fixed $\mathbf W = \{\mathbf W(p): p \in \Gamma\}$5, the standard GMC measure trivializes, and recovering nontrivial chaos would require taking $\mathbf W = \{\mathbf W(p): p \in \Gamma\}$6 to infinity as the regularization scale vanishes, which is not feasible within the GMC framework.

3. Conditional GMC Structure for Critical Continuum Random Polymers

Despite the failure of classical GMC, Clark (Clark, 2019) demonstrates that there exists a family of random measures $\mathbf W = \{\mathbf W(p): p \in \Gamma\}$7 on the path space $\mathbf W = \{\mathbf W(p): p \in \Gamma\}$8 arising as critical scaling limits of discrete directed polymers on hierarchical graphs. Key properties include:

• $\mathbf W = \{\mathbf W(p): p \in \Gamma\}$9 for all $T(p, q) = \mathbb E[\mathbf W(p)\mathbf W(q)]$0; $T(p, q) = \mathbb E[\mathbf W(p)\mathbf W(q)]$1 as $T(p, q) = \mathbb E[\mathbf W(p)\mathbf W(q)]$2.
• Second-order correlations $T(p, q) = \mathbb E[\mathbf W(p)\mathbf W(q)]$3 are mutually absolutely continuous, with

$T(p, q) = \mathbb E[\mathbf W(p)\mathbf W(q)]$4

for $T(p, q) = \mathbb E[\mathbf W(p)\mathbf W(q)]$5.

A central result is the existence, for each $T(p, q) = \mathbb E[\mathbf W(p)\mathbf W(q)]$6 and any $T(p, q) = \mathbb E[\mathbf W(p)\mathbf W(q)]$7, of a white-noise field $T(p, q) = \mathbb E[\mathbf W(p)\mathbf W(q)]$8 on $T(p, q) = \mathbb E[\mathbf W(p)\mathbf W(q)]$9 with covariance $$\mathbf M_\beta(dp) = \exp\left(\beta \mathbf W(p) - \tfrac{\beta^2}{2} T(p, p)\right) \mu(dp),$$0, such that the conditional GMC measure

$$\mathbf M_\beta(dp) = \exp\left(\beta \mathbf W(p) - \tfrac{\beta^2}{2} T(p, p)\right) \mu(dp),$$1

has law equal to $$\mathbf M_\beta(dp) = \exp\left(\beta \mathbf W(p) - \tfrac{\beta^2}{2} T(p, p)\right) \mu(dp),$$2.

In particular, the family $$\mathbf M_\beta(dp) = \exp\left(\beta \mathbf W(p) - \tfrac{\beta^2}{2} T(p, p)\right) \mu(dp),$$3 forms a cascade: $$\mathbf M_\beta(dp) = \exp\left(\beta \mathbf W(p) - \tfrac{\beta^2}{2} T(p, p)\right) \mu(dp),$$4 for any $$\mathbf M_\beta(dp) = \exp\left(\beta \mathbf W(p) - \tfrac{\beta^2}{2} T(p, p)\right) \mu(dp),$$5. This realizes the conditional GMC as a one-parameter family closed under subcritical GMC shifts with random base measure $$\mathbf M_\beta(dp) = \exp\left(\beta \mathbf W(p) - \tfrac{\beta^2}{2} T(p, p)\right) \mu(dp),$$6.

4. Technical Construction, Conditioning, and Uniqueness

The construction is carried out on an enlarged probability space realizing simultaneously $$\mathbf M_\beta(dp) = \exp\left(\beta \mathbf W(p) - \tfrac{\beta^2}{2} T(p, p)\right) \mu(dp),$$7 and an independent white noise $$\mathbf M_\beta(dp) = \exp\left(\beta \mathbf W(p) - \tfrac{\beta^2}{2} T(p, p)\right) \mu(dp),$$8 on a Hilbert space $$\mathbf M_\beta(dp) = \exp\left(\beta \mathbf W(p) - \tfrac{\beta^2}{2} T(p, p)\right) \mu(dp),$$9. The conditional field $\beta \ge 0$0 is a centered Gaussian field with covariance $\beta \ge 0$1. The conditional GMC measure

$\beta \ge 0$2

is measurable with respect to the $\beta \ge 0$3-algebra generated by $\beta \ge 0$4 and $\beta \ge 0$5. The family of conditional GMCs satisfies the following:

• Conditional expectation: $\beta \ge 0$6 for measurable $\beta \ge 0$7.
• Compatibility with the multiplicative structure under Cameron–Martin shifts required by Shamov’s axioms.
• Uniqueness in law, as they satisfy renormalization properties (I)–(IV) given in (Clark, 2019).

5. Radon–Nikodym Derivative and Interrelationship of Correlation Measures

The correlation measures between different levels of the cascade are connected via explicit Radon–Nikodym derivatives given by

$\beta \ge 0$8

Notably, the support of $\beta \ge 0$9 is typically on $\mathbf M_\beta$0, while $\mathbf M_\beta$1 almost surely places weight on $\mathbf M_\beta$2, reflecting the concentration of the polymer endpoint measure on self-intersecting path pairs due to the disorder's effect at criticality.

6. Extensions and Open Problems in Higher Dimensions

A structurally analogous conditional GMC is conjectured to govern the critical $\mathbf M_\beta$3-dimensional directed polymer and, equivalently, the critical stochastic heat equation in two spatial dimensions, as studied by Caravenna–Sun–Zygouras and Gu–Quastel–Tsai. There, a one-parameter family $\mathbf M_\beta$4 of endpoint measures is expected to satisfy

$\mathbf M_\beta$5

with $\mathbf M_\beta$6 a white noise on $\mathbf M_\beta$7 conditionally on $\mathbf M_\beta$8 and $\mathbf M_\beta$9 the renormalized local time intersection kernel. The technical challenge is the singularity of $\beta < \beta_c$0 at the diagonal and the resulting difficulty in controlling exponential moments under singular random reference measures. No complete rigorous GMC construction is currently available in this $\beta < \beta_c$1-dimensional setting, making it a central open problem in the field (Clark, 2019).

7. Significance and Broader Implications

The conditional GMC framework elucidates the mathematical structure underlying critical continuum polymer measures on fractal geometries where classical GMC fails. The existence of an intrinsic, conditional multiplicative chaos, closed under a cascade of random measure shifts, offers a canonical platform for modeling randomness in critical statistical mechanics and stochastic PDEs. The diamond fractal provides a tractable arena for establishing these conditional structures, with anticipated analogs in higher-dimensional, nonhierarchical models and in the study of critical log-correlated Gaussian fields.