Marked Condensation Phenomenon

• Marked condensation phenomenon is a sharply defined process exhibiting abrupt condensate formation with explicit diagnostic signatures across varied systems.
• In coherent Hamiltonian dynamics, it manifests as finite-time spectral collapse and dual cascades that reallocate conserved quantities.
• The phenomenon also appears in large deviation, transport-induced, optical, and deep learning contexts, offering precise markers for phase transitions.

“Marked condensation phenomenon” designates a family of condensation regimes in which the onset mechanism, structural signature, or limiting object is unusually sharp. Across the literature, the phrase is not a single standardized formalism: in coherent Hamiltonian wave dynamics it refers to a condensation process combining full phase coherence, dual cascades, exact solvability, and finite-time excitation of arbitrarily high modes; in large-deviation theory it refers to a phase transition in the mechanism realizing rare events; and in marked branching structures it denotes a local limit with a unique infinite-degree node (Biasi, 2024, Szavits-Nossan et al., 2013, Boulal et al., 23 Sep 2025). This suggests that the common denominator is not a single microscopic model, but a recurrent pattern in which condensation is accompanied by a particularly explicit diagnostic signature.

1. Conceptual scope and diagnostic signatures

In the Hamiltonian setting, the designation “marked” is tied to a very specific combination of properties: full coherence of the dynamics, a dual cascade of invariants, analytic exact solvability, and finite-time excitation of arbitrarily high modes within a deterministic conservative system (Biasi, 2024). In large-deviation problems, by contrast, the marked feature is a non-analytic change in the rate function and a switch from a many-body fluctuation mechanism to one dominated by a single degree of freedom (Szavits-Nossan et al., 2013, Szavits-Nossan et al., 2014). In marked Galton–Watson trees, the non-generic regime is explicitly identified with a “marked condensation phenomenon,” where local convergence produces a random marked tree with a node of infinite out-degree (Boulal et al., 23 Sep 2025).

The expression is also used more loosely in neighboring domains. An overview of deep-learning condensation states that “marked condensation” is not introduced there as a formal technical notion, while still describing strongly visible clustering of neurons into a small number of functional groups during nonlinear training (Xu et al., 13 Apr 2025). This heterogeneity matters: the term is best understood as a descriptive label for sharply expressed condensation, rather than as a single universally fixed definition.

2. Coherent finite-time condensation in conservative Hamiltonian systems

A particularly rigid instance appears in the resonant Hamiltonian system with complex mode amplitudes $\alpha_n(t)$, quartic resonant interactions, and Catalan-number couplings. The model preserves the action

$N=\sum_{n=0}^{\infty} |\alpha_n|^2$

and the linear energy

$E=\sum_{n=0}^{\infty} n\,|\alpha_n|^2,$

and admits a low-dimensional invariant manifold

$$\alpha_0(t)=b(t),\qquad \alpha_{n\ge 1}(t)=f_n\,c(t)\,p(t)^{\,n-1}.$$

On that manifold the infinite-dimensional dynamics reduce to coupled ODEs, and for a family of initial data the solution reaches a finite condensation time $T$ at which

$$|\alpha_n(t)|^2 \underset{t\to T}{\longrightarrow} N\,\delta_{0,n}.$$

The condensate is therefore a Kronecker delta at the fundamental mode, formed in finite time rather than as an asymptotic thermal state (Biasi, 2024).

The same solution exhibits a dual cascade. There is an inverse cascade of $N$ toward the fundamental mode and, simultaneously, a direct cascade of $E$ toward high modes. Near $t=T$, the large-$n$ tail obeys a power law

$N=\sum_{n=0}^{\infty} |\alpha_n|^2$0

after the exponential damping disappears. Hence the action collapses into $N=\sum_{n=0}^{\infty} |\alpha_n|^2$1, while the conserved energy is transferred to arbitrarily high modes. The associated Sobolev norms

$N=\sum_{n=0}^{\infty} |\alpha_n|^2$2

satisfy $N=\sum_{n=0}^{\infty} |\alpha_n|^2$3 for $N=\sum_{n=0}^{\infty} |\alpha_n|^2$4, so they diverge at finite time even though $N=\sum_{n=0}^{\infty} |\alpha_n|^2$5 remains finite by conservation (Biasi, 2024).

In physical space, with

$N=\sum_{n=0}^{\infty} |\alpha_n|^2$6

the coherent phases cause the high-frequency tail to add constructively at a distinguished point $N=\sum_{n=0}^{\infty} |\alpha_n|^2$7. Away from that point the field approaches the fundamental mode, while at that point the solution develops a jump discontinuity. The marked character of the phenomenon is therefore spectral and spatial at once: finite-time condensate formation, finite-time excitation of arbitrarily high modes, and formation of a small-scale coherent structure are analytically resolved within one deterministic Hamiltonian framework (Biasi, 2024).

3. Constraint-driven and fluctuation-driven condensation

A distinct line of work studies condensation in sums of random variables under additional global constraints. For i.i.d. nonnegative variables with a light-tailed exponential density $N=\sum_{n=0}^{\infty} |\alpha_n|^2$8, conditioning simultaneously on the sample mean $N=\sum_{n=0}^{\infty} |\alpha_n|^2$9 and the quadratic statistic $E=\sum_{n=0}^{\infty} n\,|\alpha_n|^2,$0 with $E=\sum_{n=0}^{\infty} n\,|\alpha_n|^2,$1 produces a phase diagram with forbidden region $E=\sum_{n=0}^{\infty} n\,|\alpha_n|^2,$2, fluid region $E=\sum_{n=0}^{\infty} n\,|\alpha_n|^2,$3, and condensed region $E=\sum_{n=0}^{\infty} n\,|\alpha_n|^2,$4. In the condensed phase, one variable has $E=\sum_{n=0}^{\infty} n\,|\alpha_n|^2,$5, the saddle-point solution breaks down, and the conditional probability changes from exponential large-deviation scaling to a sub-exponential form proportional to

$E=\sum_{n=0}^{\infty} n\,|\alpha_n|^2,$6

The marked signature is precisely the change of scaling and the corresponding non-analyticity in the effective rate function (Szavits-Nossan et al., 2013).

The same mechanism was generalized to joint large deviations of

$E=\sum_{n=0}^{\infty} n\,|\alpha_n|^2,$7

For $E=\sum_{n=0}^{\infty} n\,|\alpha_n|^2,$8, the critical line is

$E=\sum_{n=0}^{\infty} n\,|\alpha_n|^2,$9

and for $$\alpha_0(t)=b(t),\qquad \alpha_{n\ge 1}(t)=f_n\,c(t)\,p(t)^{\,n-1}.$$0 the condensed contribution has size

$$\alpha_0(t)=b(t),\qquad \alpha_{n\ge 1}(t)=f_n\,c(t)\,p(t)^{\,n-1}.$$1

Here the condensate is macroscopic in the constrained statistic $$\alpha_0(t)=b(t),\qquad \alpha_{n\ge 1}(t)=f_n\,c(t)\,p(t)^{\,n-1}.$$2, even when it is only subextensive in the bare mass $$\alpha_0(t)=b(t),\qquad \alpha_{n\ge 1}(t)=f_n\,c(t)\,p(t)^{\,n-1}.$$3. This sharpens the notion of a marked condensate: the singled-out degree of freedom is selected by the geometry of the constrained large-deviation problem rather than solely by the tail of the underlying distribution (Szavits-Nossan et al., 2014).

A related classification uses the participation ratio

$$\alpha_0(t)=b(t),\qquad \alpha_{n\ge 1}(t)=f_n\,c(t)\,p(t)^{\,n-1}.$$4

for power-law variables under a superextensive mass constraint $$\alpha_0(t)=b(t),\qquad \alpha_{n\ge 1}(t)=f_n\,c(t)\,p(t)^{\,n-1}.$$5. For $$\alpha_0(t)=b(t),\qquad \alpha_{n\ge 1}(t)=f_n\,c(t)\,p(t)^{\,n-1}.$$6, the phase boundary is

$$\alpha_0(t)=b(t),\qquad \alpha_{n\ge 1}(t)=f_n\,c(t)\,p(t)^{\,n-1}.$$7

The regime $$\alpha_0(t)=b(t),\qquad \alpha_{n\ge 1}(t)=f_n\,c(t)\,p(t)^{\,n-1}.$$8 is condensed, $$\alpha_0(t)=b(t),\qquad \alpha_{n\ge 1}(t)=f_n\,c(t)\,p(t)^{\,n-1}.$$9 is homogeneous, and the line $T$0 supports a weak condensation phenomenon in which $T$1 remains nonzero but the condensate bump can have non-vanishing relative width (Gradenigo et al., 2017).

A complementary formulation is the condensation of fluctuations. In the Gaussian model, the probability of a macrovariable $T$2 has a large-deviation form $T$3, while an exponentially biased ensemble

$T$4

induces a mean-field-like effective interaction. Condensation then occurs on average in the biased ensemble and, by duality, as a rare-event structure in the original ensemble. This establishes that condensation of fluctuations may occur even in a non-interacting system (Zannetti et al., 2014).

4. Transport-induced real-space condensation

Real-space condensation can also be induced by coupled transport processes. In the C2C model derived from the Discrete Nonlinear Schrödinger equation, the site masses $T$5 conserve

$T$6

with mass density $T$7 and energy density $T$8. The equilibrium condensation threshold is the critical parabola

$T$9

For $$|\alpha_n(t)|^2 \underset{t\to T}{\longrightarrow} N\,\delta_{0,n}.$$0 the state is homogeneous; for $$|\alpha_n(t)|^2 \underset{t\to T}{\longrightarrow} N\,\delta_{0,n}.$$1 a finite fraction of the energy localizes on one site (Gotti et al., 2022).

The marked feature of the open nonequilibrium problem is that both reservoirs can impose subcritical boundary conditions, while asymmetric driving still creates an internal supercritical region. In the unpinned and pinned variants, the parametric bulk profile $$|\alpha_n(t)|^2 \underset{t\to T}{\longrightarrow} N\,\delta_{0,n}.$$2 crosses the line $$|\alpha_n(t)|^2 \underset{t\to T}{\longrightarrow} N\,\delta_{0,n}.$$3, so condensation appears in the interior of the chain rather than at a boundary. The mechanism relies on two conservation laws and two coupled currents; the paper stresses that such internal condensation is absent in the comparable single-conservation-law settings discussed there (Gotti et al., 2022).

Pinning changes the macrodynamics. In the pinned dynamics, a condensate remains fixed on one site and absorbs energy indefinitely, so no stationary state exists. In the unpinned dynamics, peaks can move, decay in subcritical regions, and reform in the supercritical interval, producing a stationary nonequilibrium state with diffusive currents

$$|\alpha_n(t)|^2 \underset{t\to T}{\longrightarrow} N\,\delta_{0,n}.$$4

This provides a marked transport-induced condensation mechanism: localization is generated, positioned, and dynamically regulated by coupled mass-energy transport under nonequilibrium forcing (Gotti et al., 2022).

5. Optical, photonic, and spectral condensation

In linear photonic mode systems with noise, classical light condensation is formulated in loss-gain space rather than photon-energy space. The mode amplitudes obey

$$|\alpha_n(t)|^2 \underset{t\to T}{\longrightarrow} N\,\delta_{0,n}.$$5

with noise correlations

$$|\alpha_n(t)|^2 \underset{t\to T}{\longrightarrow} N\,\delta_{0,n}.$$6

The steady-state occupancy is

$$|\alpha_n(t)|^2 \underset{t\to T}{\longrightarrow} N\,\delta_{0,n}.$$7

and the effective chemical potential is $$|\alpha_n(t)|^2 \underset{t\to T}{\longrightarrow} N\,\delta_{0,n}.$$8. Condensation occurs when the spectral filtering near the lowest-loss mode has exponent $$|\alpha_n(t)|^2 \underset{t\to T}{\longrightarrow} N\,\delta_{0,n}.$$9; then the higher-loss modes have finite capacity, and once the total power exceeds

$N$0

all additional power flows into the lowest-loss mode. The sharp transition from multi-mode to single-mode oscillation is the marked signature. The same work emphasizes that this phenomenon is classical and loss-based, not necessarily genuine photon Bose–Einstein condensation (Fischer et al., 2012).

In graded-index multimode fibers, weak structural disorder dramatically accelerates thermalization and wave condensation. The nonequilibrium evolution of the modal populations $N$1 is described by a discrete kinetic equation with Rayleigh–Jeans equilibrium

$N$2

and condensation into the fundamental mode occurs when $N$3. The thermalization-length reduction obeys

$N$4

so weak disorder accelerates condensation by several orders of magnitude. Experiments reported condensate fractions of up to $N$5 in the fundamental mode, providing a statistical explanation of optical beam self-cleaning (Fusaro et al., 2020).

A broader nonequilibrium analogue is spectral condensation, defined as the redistribution of broadband spectral power into a dominant mode. For the power spectrum $N$6, with dominant peak $N$7, the paper introduces spectral measures

$N$8

based on moments of the normalized spectrum around $N$9, and finds inverse power-law relations

$E$0

across thermoacoustic, aeroacoustic, aeroelastic, optical, and electronic systems. Representative exponents are $E$1 for $E$2, $E$3 for $E$4, and $E$5 for $E$6. Here the marked character is the sharp transition from broadband spectra to a narrow dominant peak and the emergence of universal scaling across disparate systems (Pavithran et al., 2020).

6. Condensation in deep learning

In deep learning, condensation denotes the tendency of neurons in the same layer to form groups with similar outputs during nonlinear training. For a two-layer network

$E$7

a rigorous phase diagram is obtained by introducing the scaling exponents

$E$8

with $E$9 and $t=T$0. The condensed regime is

$t=T$1

while the linear regime is $t=T$2 or $t=T$3. In the condensed regime, the relative displacement of hidden weights diverges on an initial time interval and their norm aligns with the data direction

$t=T$4

so that

$t=T$5

This is a marked condensation in the literal sense of overwhelming directional concentration (Chen et al., 2023).

An overview of the field extends the picture to CNNs, ResNets, and Transformers. It states that during nonlinear training neurons in the same layer tend to condense into groups with similar outputs, that the number of condensed clusters in the same layer typically increases monotonically as training progresses, and that small weight initializations or Dropout optimization can facilitate this process. The same overview also notes that “marked condensation phenomenon” is not a formal concept there; rather, strong block structure in cosine-similarity heatmaps or tight clustering in feature space provides the empirical diagnostic. It further associates condensation with better generalization and reports a correlation with stronger reasoning abilities in transformer-based LLMs (Xu et al., 13 Apr 2025).

7. Marked trees, thermodynamic criteria, and comparative distinctions

For marked Galton–Watson trees, the conditioning variable is the number of marked vertices

$t=T$6

If the offspring law is critical, or subcritical but generic after the $t=T$7-tilt construction, conditioning on $t=T$8 with $t=T$9 yields local convergence to a marked Kesten’s tree with an infinite spine. In the non-generic subcritical case, the local limit is instead a marked condensation tree with exactly one vertex of infinite degree. The paper identifies this non-generic regime with the marked condensation phenomenon. The distinction is structural: the generic phase spreads marks along an infinite spine, whereas the non-generic phase concentrates the mark burden around a single condensed hub (Boulal et al., 23 Sep 2025).

A thermodynamic counterpart appears in the phase diagram of quantum fluids, where the transition from a normal fluid to a quantum fluid is identified with the change of sign of the chemical potential. In that framework, $n$0 characterizes normal phases, $n$1 marks the onset of condensation, and $n$2 characterizes the quantum fluid phase. Condensation is defined there as a macroscopic number of particles occupying their own many-body ground state, encompassing ideal Bose–Einstein condensation and what the authors call “Fermi-Dirac condensation” (Romero-Rochin, 2011).

Taken together, these works establish several objective distinctions. Condensation may occur on average or in fluctuations; it may be coherent or kinetic; it may be spectral, real-space, combinatorial, or representational; and it may or may not require interactions in the original ensemble. What makes a condensation phenomenon “marked” in the surveyed literature is the presence of a sharply identifiable hallmark: finite-time spectral collapse, a non-analytic rate-function change, internally generated localization under subcritical boundaries, a universal inverse-power-law spectral signature, near-complete neuron alignment, or a local limit with one infinite-degree node (Biasi, 2024, Zannetti et al., 2014).