# Statistical Mechanics of Exponentially Many Low Lying States

Swapnamay Mondal1,2
$~{}^{1}$ Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin,
Ireland
$~{}^{2}$ School of Mathematics, Trinity College, Dublin 2, Ireland
<EMAIL_ADDRESS>
###### Abstract:

It has recently been argued that for near-extremal black holes, the entropy
and the energy above extremality respectively receive a $logT$ and a
$T$-linear correction, where $T$ is the temperature. We show that both these
features can be derived in a “low but not too low” temperature regime, by
assuming the existence of exponentially many low lying states, without using
any specific theory.

##  1 Introduction

Microscopic explanation of the entropy of non-extremal black holes has
received a fair degree of attention [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16]. However, due to non-zero temperature, there is more to the
thermodynamics of such black holes than just entropy. E.g. semi-classical
analysis entails that slightly above extremality, energy or mass of such black
holes grows quadratically with temperature. The proportionality constant,
which has the dimension of inverse mass, gives a mass scale $M_{gap}$. It has
been suggested that below this scale, Hawking’s analysis may not be trustable
[17, 18, 19]. Microscopic descriptions, such as [20, 21] indeed exhibit a
gapped energy spectrum. But these descriptions are limited to supersymmetric
black holes.

Recently there has been a surge of interest in near-extremal black holes [22,
23, 24, 25, 26, 27, 28]. In particular, careful consideration of quantum
effects suggests the absence of a gap for near-extremal Reissner-Nordström
black holes [29]. This comes from certain boundary modes, captured by the
Schwarzian theory [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44]. This sector changes the low temperature thermodynamics significantly, in
particular giving rise to a $log\,T$ dependence in entropy and T-linear term
in energy, where $T$ is the temperature.

A microscopic understanding of these formulae is lacking to the best of our
knowledge. One immediate problem in this regard is that for finite
temperature, one would require information about the spectrum (i.e. not just
the ground states), which would change considerably as the system is changed
from the D-brane regime to the black hole regime. For conformal field theory
descriptions, it has been suggested that $T\bar{T}$ deformation captures such
a change [45].

Anyhow, if extremality corresponds to exponentially many ground states (which
has been argued against for non-supersymmetric case [29]), it is natural to
wonder whether the nearby situation of exponentially many low lying states
could explain some of the findings for near-extremal black holes, at least
qualitatively.

One becomes hopeful upon noting that the Schwarzian theory also captures the
low energy physics of the Sachdev-Ye-Kitaev model or SYK model [30, 31, 32,
46], for which the energy gap between low lying states falls off exponentially
in the system size, leading to exponentially many low lying states. This in
particular implies the lack of quasi particle excitations, indicating strange
metallic behaviour.

Motivated by this observations, we consider a system with $e^{S_{0}}$ low
lying states distributed over an energy interval $\Delta$, and separated from
rest of the spectrum by a gap $E_{gap}\gg\Delta$. We analyze the statistical
mechanics of this system in different low temperature ($T\ll E_{gap}$) regimes
and show that in a “low but not too low” temperature regime, qualitative
features of Schwarzian thermodynamics, such as $log\,T$ dependence in entropy
and $T$-linear dependence of energy, are readily retrieved.

In section 2 we survey systems with exponentially many low lying states. In
section 3 we analyze the statistical mechanics of such systems, under some
simplifying assumptions. In section 4, we discuss implications of our
findings.

##  2 Systems with large number of low lying states

In this section we quickly review some physical systems, which exhibit or are
likely to exhibit exponentially many low lying states.

### Near-extremal black holes:

Extremal black holes carry zero temperature and hence their entropy is the
logarithm of the ground state degeneracy. It is unclear though what protects
this degeneracy. It has been argued that exponential of the extremal entropy
actually counts not only the ground states but all the states below a certain
gap scale [19]. Anyhow, technically extremal black holes are easier to deal
with owing to the decoupling of the near-horizon region.

For non-extremal black holes, the near horizon region does not decouple. It is
useful to separate such a spacetime into a Near Horizon Region (NHR) and a Far
Away Region (FAR) [28]. For the near extremal case, the NHR region still
retains some flavours of the $AdS_{2}\times S^{2}$ space. After reduction over
transverse $S^{2}$, one gets an effective theory on the two dimensional space
spanned by the radial and temporal coordinates. This theory in particular
incorporates Jackiw Teitelboim (JT) gravity [47, 48], which appears abundantly
in the context of near-extremal black holes [49, 50, 51, 52, 53, 54, 55, 56,
57, 58, 59].

Slow variation of transverse $S^{2}$ in NHR breaks the $AdS_{2}$ isometries.
This symmetry breaking scale has been identified with the gap scale $M_{gap}$
[50]. The low temperature dynamics is governed by the (pseudo)Goldstones of
this symmetry breaking [33]. For more on Kaluza Klein reduction in this
settings, see [60, 61]. After careful consideration of one loop correction of
the JT mode, one obtains the following expression for entropy and energy
(above extremality) [29]

$\displaystyle{}S$
$\displaystyle=S_{0}+\frac{4\pi^{2}\Phi_{b,Q}}{\beta}-\frac{3}{2}\log{}\frac{\beta}{3\Phi_{b,Q}}$
$\displaystyle E$
$\displaystyle=\frac{2\pi^{2}\Phi_{b,Q}}{\beta^{2}}+\frac{3}{2\beta}\,,$ (1)

where $\Phi_{b,Q}\sim M_{gap}^{-1}$ is the value of the dilaton at fixed
charge $Q$ and $\beta$ is the inverse temperature. For very low temperatures,
energy above extremality scales linearly with temperature, thus lacking any
characteristic energy scale and hence suggesting a gapless spectrum. Since
this follows from a gravity analysis, rather than a microscopic one, the
gapless spectrum is better thought of as a smeared version of densely spaced
low lying states. It might seem at odds with vanishing density of states near
the ground state, as suggested by $\lim_{T\rightarrow 0}e^{S}=0$. However
there is no contradiction, as the “small” density of states can still be large
in system size. In fact, this possibility is explicitly realized for SYK
model, as we shall see latter in this section.

### Brane systems:

Microscopic descriptions of black holes are often constructed in terms of an
effective string, wrapping a circle of size $R$ [62, 63]. For example, for D4
brane black holes, this string represents an M5 brane reduced over a four-
cycle inside the Calabi-Yau threeefold, and wraps the M-theory circle. Charges
carried by the black hole appear as the conserved charges in the worldvolume
theory on this effective string. Hence black hole microstate counting amounts
to counting states in the worldvolume theory in a given charge sector. To
facilitate this task, one goes to a regime, where the system is well described
by a free conformal field theory (CFT), where central charge can be determined
solely from the field content. Then one uses Cardy’s formula to count the
states [64]. Such a CFT captures a family of black holes at one go, including
non-extremal ones. In fact near-extremal black hole entropies have been
successfully reproduced from such state counting in a number of cases [3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].

Note that it is not that one writes down a worldvolume Lagrangian and it
happens to describe a CFT (as is the case for D3 branes). Rather, one simply
looks at the massless field content and assumes that in an appropriate regime
their interaction can be ignored, which is sometimes justified by dilute
nature of the system at low energies [3]. In some instances, a sigma model
description of the effective string dynamics also exists [65].

Anyhow, the worldvolume theory has no reason to be conformal to start with and
the CFT description is valid only in a particular corner of the moduli
space111For D4-brane black holes, this CFT limit point can be rephrased as a
decoupling limit [66, 45].. Away from the conformal point, one expects the
degeneracy to be protected for the supersymmetric states, but not for the non-
supersymmetric ones. The degenerate non-supersymmetric states of the CFT are
likely to split, as one moves away from the CFT point. For near-supersymmetric
states, the splitting is likely to be small, leading to large number of
closely spaced states.

Such near-supersymmetric states can sometimes be associated with a brane-
antibrane system [5, 9, 11]. Should one consider the low energy theory
describing the specific brane-antibrane system, it is conceivable that the
near-extremal states would appear as low energy states in this theory.

A particularly curious case is that of extremal non-supersymmetric states.
Since these are not even nearly supersymmetric, corresponding states need not
remain even nearly degenerate as one moves away from the CFT point. On the
other hand semi-classical gravity would have us believe that this state is
highly degenerate. To resolve this tension a study the quantum mechanics of
such systems is required.

Lastly, note that the moving away from CFT point discussed here is not of the
sort discussed in [45], which captures gravitational effects and does not
affect the degeneracies.

### Sachdev-Ye-Kitaev model:

SYK model [30, 31, 32, 46] is a model of disordered Majorana fermions. It’s
claims to fame are manifold. It is solvable in a specific large N limit and
saturates the chaos bound222The bound on chaos has been related to Einstein
velocity bound [67, 68]. [69]. In the same regime, it develops an emergent
reparameterization symmetry, which is both spontaneously and explicitly
broken, resulting in (pseudo)Goldstones, which dominate the low energy
physics. This pattern of symmetry breaking is reminiscent of near $AdS_{2}$
spaces [33].

One key feature of SYK model is it’s lack of quasiparticles. This feature is
particularly interesting in connection with strange metals. The lack of quasi
particles manifests itself in exponentially small (in system size) level
spacing of low lying states. In other words there are exponentially many low
lying states. This leads to a non-vanishing zero temperature entropy, provided
one take the large $N$ limit first and zero temperature limit afterwards. If
these limits are taken in the opposite order, one obtains a vanishing entropy
density. This means there is no ground state degeneracy, rather there are
exponentially many low lying states. An alternate way to think of the non-
commutativity of the limits is that they capture the system at different low
temperature regimes: taking the system size to infinity first and
$T\rightarrow 0$ limit afterwards, corresponds to probing the system at a
temperature that is low but not low enough to resolve the closely spaced low
lying states. This “low but not too low” regime would be important for the
next section. The other order of limits corresponds to temperatures that are
low compared to all other scales.

It is instructive to look at the density of states [70, 71, 72, 35]

$\displaystyle D(E)\sim\frac{1}{N}e^{Ns_{0}}\sinh\sqrt{2N\gamma E}\,,$ (2)

where the constants $s_{0},\gamma$ are not particularly relevant for us. The
factor $e^{Ns_{0}}$ ensures the existence of an extensive entropy. Note, the
degeneracy is exponentially large in system size and at the same time it
vanishes at low energies, just like near-extremal black holes discussed below
(1).

##  3 Statistical mechanics of exponentially many low lying states

In SYK model, there is no clean separation between these low lying states,
contributing to zero temperature entropy and the rest of the states, as
reflected in the continuous density of states (2). However, to distill the
effect of the low lying states, we make the simplifying assumption that such
states are separated by a gap from the rest of the spectrum. Such spectra
might arise in microscopic description of non-supersymmetric extremal/near-
extremal black holes, along the lines of [20, 21]. We shall not attempt it
here and simply treat such a spectrum as a toy model.

Let us start with a system, with $\Omega:=e^{S_{0}}$ degenerate ground states
and a gap $E_{gap}$ in the spectrum. Now we slightly perturb the system, such
that the ground state degeneracy is lifted leading to $e^{S_{0}}$ states
reaching upto energy $\Delta\ll E_{gap}$. In that case, for temperatures $T\ll
E_{gap}$, the contribution of states with $E\geq E_{gap}$ can be ignored in
the partition function and we can approximate

$\displaystyle Z$ $\displaystyle\simeq\sum_{E\leq\Delta}d(E)e^{-\beta E}\,.$
(3)

For the sake of simplicity, we further assume even spacing for the degeneracy
lifted states, i.e. we assume
$d(E)=\sum_{n=1}^{\Omega}\delta(E-n\Delta/\Omega)$. With this assumption, the
partition function can be written in closed form:

$\displaystyle Z$
$\displaystyle=\frac{1-e^{-\Delta\beta}}{e^{\Delta\beta/\Omega}-1}\,.$ (4)

Since there are now two relevant energy scales, namely $\Delta$ and
$\Delta/\Omega$, there are three low temperature regimes to be considered, as
discussed in the following.

  1. 1.

Moderately low temperatures $(E_{gap}\gg T\gg\Delta)$: In this regime,
$\Delta\beta\ll 1$. Hence

$\displaystyle{}\ln{}Z$ $\displaystyle\simeq
S_{0}-\frac{1}{2}(1+\Omega^{-1})\Delta\beta+\frac{1}{24}(1-\Omega^{-2})\Delta^{2}\beta^{2}+\dots\,$
$\displaystyle{}U$
$\displaystyle=-\partial_{\beta}\ln{}Z\simeq\frac{1}{2}(1+1/\Omega)\Delta-\frac{1}{12}(1-1/\Omega^{2})\Delta^{2}\beta+\dots$
$\displaystyle S$ $\displaystyle=\beta U+\ln{}Z\simeq
S_{0}-\frac{1}{24}(1-1/\Omega^{2})\Delta^{2}\beta^{2}+\dots\,.$ (5)

We note (5) admits smooth $\Delta\rightarrow 0$ limit, as well as a “high
temperature limit” $\Delta\beta\rightarrow 0$. Both have the entropy $S_{0}$,
but the $\Delta\beta\rightarrow 0$ limit has a non-vanishing internal energy
as well.

  2. 2.

Low but not too low temperatures $(\Delta\gg T\gg\Delta/\Omega)$: In this
regime, $\Delta\beta\gg 1$, but $\Delta\beta/\Omega\ll 1$. Hence

$\displaystyle{}\ln{}Z$ $\displaystyle\simeq
S_{0}-\ln{}\beta\Delta-\frac{\beta\Delta}{2\Omega}+\dots$ $\displaystyle{}U$
$\displaystyle\simeq T+\frac{\Delta}{2\Omega}+\dots$ $\displaystyle S$
$\displaystyle\simeq S_{0}+\ln{}\frac{T}{\Delta}+\dots\,.$ (6)

If we take large system size limit first, we get a non-vanishing zero
temperature entropy. Whereas naive extrapolations to $T\rightarrow 0$ limit
gives vanishing $e^{S}$ and a finite energy.

  3. 3.

Very low temperatures $(T<<\Delta/\Omega)$: In this regime,
$\beta\Delta/\Omega\gg 1$. Hence

$\displaystyle{}\ln{}Z$
$\displaystyle=-\frac{\Delta\beta}{\Omega}+e^{-\frac{\Delta\beta}{\Omega}}+\dots$
$\displaystyle{}U$
$\displaystyle\simeq\frac{\Delta}{\Omega}+\frac{\Delta}{\Omega}e^{-\frac{\Delta\beta}{\Omega}}+\dots$
$\displaystyle S$
$\displaystyle\simeq\frac{\beta\Delta}{\Omega}e^{-\frac{\beta\Delta}{\Omega}}+\dots\,.$
(7)

In zero temperature limit, the energy approached that of the true ground state
and the entropy vanishes.

The “low but not too low” temperature regime $\Delta\gg T\gg\Delta/\Omega$ is
of particular interest. Firstly we note that a similar temperature regime
exists for the SYK model333We thank Gustavo Turiaci for discussions on this
issue. as discussed towards the end of the last section. The temperature in
SYK model should be low in order to access the deep infrared, but not so low
that the exponentially small energy gap over the ground state becomes evident
and the true ground state reigns supreme.

This temperature regime also includes the natural temperature444Ref. [73]
assigned temperature to near-extremal CFT states in a similar fashion.
associated with the low lying states in the energy interval
$(\Delta/\Omega,\Delta)$,

$\displaystyle T^{-1}_{natural}$ $\displaystyle:=\frac{\Delta S}{\Delta
E}\simeq\frac{\ln{}\Omega}{\Delta}\,.$ (8)

For large $\Omega$, which is the case for big black holes, the condition
$\Delta\gg T_{natural}\gg\Delta/\Omega$ is readily satisfied. The derivation
of (6) assumes a hierarchy between the energy scales, $\Delta/\Omega$ and
$\Delta$. This assumption is clearly justified for large black holes. But even
for small charges, where the black hole picture is not trustworthy, $\Omega$
can be large enough to allow for a “low but not too low” temperature regime.
E.g. for 1/8 BPS black holes, the smallest values of $\Omega$ are
$12,56,208,684,\dots$.

This regime’s claim to fame is that up to numerical coefficients, (6) has the
same behaviour as near-extremal thermodynamics (1) for low temperatures.
Comparison of the $\log{}T$ term in entropies of (6) and (1) reveals
$\Delta\sim M_{gap}$. The numerical mismatches are hardly unexpected, since
the density of states for near-extremal black holes scale as
$\rho(E)\sim\sqrt{E}$ for very small energies (see (2)), whereas we are
working with a constant $\rho(E)$ for low energies. It is then remarkable that
such a simple picture nevertheless succeeds in capturing the key features. We
believe, this signifies that the statistical mechanics of near-extremal black
holes is essentially that of exponentially many low lying states. In
particular, this serves as a useful guideline for modelling near-extremal
black holes.

A much debated issue for near-extremal black holes is the presence or absence
of a gap. Recent findings seem to suggest absence of a gap [29]. However, the
methods used to reach this conclusion are not refined enough to resolve
individual microstates. Thus the suggested gapless spectrum might be an
approximation for densely packed low lying states. We note that the low energy
spectrum considered in this paper is such a spectrum. At low enough energy
resolutions, the spectrum at hand would seem gapless (below energy $\Delta$),
whereas at higher resolutions, individual microstates are revealed.

The fact that (6) captures the thermodynamic features of only the Schwarzian
part, is significant. The Schwarzian dynamics emanates from the boundary of
near$AdS_{2}$ space [33]. If one deforms the black hole towards extremality,
the throat becomes deeper and eventually the near horizon $AdS_{2}$ decouples.
On the $AdS_{2}$ boundary lives the dual $CFT_{1}$ which in turn is obtained
by letting the microscopic D-brane quantum mechanics flow to deep infrared,
where it captures only the ground states. This suggests that the states of the
Schwarzian theory are closely related to the states of the $CFT_{1}$, i.e. the
degenerate ground states of the D-brane quantum mechanics. Recovery of
qualitative features of the Schwarzian thermodynamics from a system of near-
degenerate low lying states, then seems to suggest that these low lying states
correspond to the ground states of the corresponding extremal black hole.

There is however one key difference with the near-extremal black hole set up
of [29]. Equation (6), which entails $e^{S}\sim T$, does not imply a vanishing
$e^{S}$ at zero temperature, as it is derived for “low but not too low”
temperatures. This is unlike the results of [29], which are supposed to be
valid even for very low temperatures. One response to this mismatch could be
that the spectrum considered in this paper fails to capture this particular
aspect of near-extremal physics. Another possibility, which we would like to
consider, is to allow room for reasonable doubt regarding the applicability of
a gravity analysis of near-extremal black holes, to arbitrarily low
temperatures.

##  4 Discussion

We have shown that up to numerical coefficients, the low energy Schwarzian
thermodynamics can be explained by a simple ansatz hypothesizing exponentially
many low lying states, separated by a gap from rest of the spectrum. This in
particular suggests that low energy states of the Schwarzian theory are none
other than the states of the corresponding extremal black hole, albeit
degeneracy lifted, the lifting scale being proportional to the $AdS_{2}$
isometry breaking scale.

If the low energy Schwarzian part arises from the near-degenerate low lying
states, where does the rest of near-extremal thermodynamics come from? The
natural answer seems to be the rest of the spectrum. To be more precise, for
temperatures well above the splitting scale $\Delta$, the thermodynamics of
the system should be well explained by the excited states above the near-
degenerate low lying states. Although explicit descriptions of D-brane quantum
mechanics are readily available (e.g. [20, 21]), they are likely to be
inadequate for checking this, as the spectrum would be significantly modified
as one moves from D-brane regime to the black hole regime.

Another intriguing possibility hinted at by our findings is realization of a
$nCFT_{1}$ as low energy theory of a D-brane system with exponentially many
low lying states. For extremal black holes, at least for supersymmetric ones,
$AdS_{2}/CFT_{1}$ correspondence is realized as follows [74]. The somewhat
trivial $CFT_{1}$ arises as the infrared fixed point of the gapped D-brane
quantum mechanics containing exponentially many ground states (e.g. [20, 21]),
whereas the dual $AdS_{2}$ arises as the near horizon geometry of the
corresponding black hole. For near-extremal black holes, the NHR region leads
to a $nAdS_{2}$ space and one has a $nAdS_{2}/nCFT_{1}$ correspondence [33,
75]. It is natural to wonder whether the $nCFT_{1}$ can arise as low energy
effective theory of a D-brane quantum mechanics. Our findings emboldens this
expectation, and suggests such a D-brane quantum mechanics should have
exponentially many low lying states. Note, that such a D-brane quantum
mechanics is conceivable even for extremal non-supersymmetric states.

Perhaps the most intriguing implication of this work is for strange metals.
Strange metals have long been related to black holes [76, 77]. For example,
the Planckian timescale [78, 79], which contrrols the electronic dynamics in
cuprate strange metals [79], resembles the ring down timescale of black holes
[80, 81]. One way to realize Planckian transport has been based on models
without quasi particles [67, 82], or equivalently as systems with
exponentially many low lying states. Our work suggests the possibility that
these states might be realized as degeneracy lifting of a highly degenerate
ground state, with degeneracy protected by some symmetry (possibly emergent).
Emergent supersymmetry in disordered systems might be one such symmetry [83].

### Acknowledgement:

We would like to thank Gustavo Turiaci for useful discussions. This work was
supported by Dublin Institute for Advanced Studies.

##  Appendix A Exponentially many low lying states from supersymmetric quantum
mechanics

Here we provide an explicit realization of exponentially many low lying states
by soft degeneracy lifting of degenerate ground states. Exact degeneracy is a
rather fine tuned situation, and is supposed to be destroyed unless protected
by some symmetry. Such a symmetry could be supersymmetry. Supersymmetry
algebra demands that a supersymmetric ground state must have exactly zero
energy. So if there are multiple supersymmetric ground states, their
degeneracy (index to be precise) is protected by supersymmetry. Such systems
naturally arise as microscopic description of supersymmetric black holes [20,
21]. This system is gapped and in the deep infrared, only the ground states
survive and constitute a somewhat trivial $CFT_{1}$, holographically dual to
the near horizon $AdS_{2}$ [74]. If the supersymmetry is softly broken, the
ground state degeneracy would be lifted leading to exponentially many low
lying states. A string theoretic realization of such a situation is not
inconceivable, although we do not attempt it here.

To demonstrate the soft degeneracy lifting, it suffices to consider a bunch of
$\mathcal{N}=1,d=4$ chiral multiplets, captured by a Lagrangian

$\displaystyle L$
$\displaystyle=\partial_{\mu}\phi_{i}^{\dagger}\partial^{\mu}\phi_{i}-i\bar{\psi}_{i}\bar{\sigma}^{\mu}\partial_{\mu}\psi_{i}-\frac{1}{2}\frac{\partial^{2}W}{\partial\phi_{i}\partial\phi_{j}}\psi_{i}\psi_{j}-\frac{1}{2}\frac{\partial^{2}\overline{W}}{\partial\phi_{i}^{\dagger}\partial\phi_{j}^{\dagger}}\bar{\psi}_{i}\bar{\psi}_{j}-\left(\frac{\partial
W}{\partial\phi_{i}}\right)\left(\frac{\partial
W}{\partial\phi_{i}}\right)^{\dagger}\,.$ (9)

Here $(\phi_{i},\psi_{i})$ respectively are the complex scalar and Weyl spinor
in $i^{th}$ chiral multiplet, and $W(\phi)$ is the superpotential.

Let $\phi_{0}:=\\{\phi_{0,i}\\}$ be an extremum of the superpotential and
hence a classical supersymmetric vacuum of the theory. Around the extremum, we
can expand $\phi_{i}=\phi_{0,i}+\varphi_{i}$. The zero mode dynamics around
any extremum is captured by the Lagrangian with disjoint bosonic and fermionic
dynamics

$\displaystyle L$
$\displaystyle=\dot{\varphi}_{i}^{\dagger}\dot{\varphi}_{i}-i\bar{\psi}_{i}\dot{\psi}_{i}-\frac{1}{2}M_{ij}\psi_{i}\psi_{j}-\frac{1}{2}\overline{M}_{ij}\bar{\psi}_{i}\bar{\psi}_{j}-M_{ij}\overline{M}_{ik}\varphi_{j}\overline{\varphi}_{k}+\dots\,,$
(10)

where
$M_{ij}=\frac{\partial^{2}W}{\partial\phi_{i}\partial\phi_{j}}\Big{|}_{\phi=\phi_{0}}$.
Existence of multiple classical minima then corresponds to multiple quantum
ground states with zero energy. A soft breaking of supersymmetry would lift
this degeneracy, yielding many closely spaced low lying states. In the
following we demonstrate the essential point for a single chiral multiplet.

### A single chiral multiplet:

For a single chiral multiplet $(\phi,\psi)$, the superpotential $W(\phi)$ can
still admit multiple minima. Around each such vacuum, the Lagrangian could be
expanded as

$\displaystyle L$
$\displaystyle=\dot{\varphi}^{\dagger}\dot{\varphi}-i\bar{\psi}\dot{\psi}-\frac{1}{2}M\psi\psi-\frac{1}{2}\overline{M}\bar{\psi}\bar{\psi}-|M|^{2}|\varphi|^{2}+\dots\,,$
(11)

with $M=\frac{\partial^{2}W}{\partial\phi^{2}}\Big{|}_{\phi=\phi_{0}}$.

Expanding $\varphi=\frac{1}{\sqrt{2}}(x+iy)$, the bosonic part is seen to be a
compound harmonic oscillator with energy levels

$\displaystyle E_{n_{x},n_{y}}$ $\displaystyle=(n_{x}+n_{y}+1)|M|\,,\quad
n_{x},n_{y}=0,1,2,\dots\,.$ (12)

Coming to the fermionic part, we first recall (using the convention of [84])

$\displaystyle\psi\psi$
$\displaystyle=\psi^{\alpha}\psi_{\alpha}=-2\psi^{1}\psi^{2}\,,\bar{\psi}\bar{\psi}=2\bar{\psi}_{1}\bar{\psi}_{2}=2\bar{\psi}^{1}\bar{\psi}^{2}\,,\,\bar{\psi}\psi=\bar{\psi}^{1}\psi^{1}+\bar{\psi}^{2}\psi^{2}\,.$
(13)

The fermionic Hamiltonian is

$\displaystyle H_{F}$
$\displaystyle=-M\psi^{1}\psi^{2}-M^{*}\bar{\psi}^{1}\bar{\psi}^{2}\,.$ (14)

Taking the vacuum $|0\rangle$ to be annihilated by $\psi^{1},\bar{\psi}^{2}$,
we have

$\displaystyle H_{F}|0\rangle$
$\displaystyle=0\,,\,H_{F}\psi^{2}\bar{\psi}^{1}|0\rangle=0\,,\,H_{F}\bar{\psi}^{1}|0\rangle=M\psi^{2}|0\rangle\,,\,H_{F}\psi^{2}|0\rangle=M^{*}\bar{\psi}^{1}|0\rangle\,.$
(15)

So the eigenvalues of $H_{F}$ are $0,0,\pm|M|$, implying the fermionic ground
state energy to be $-|M|$, which exactly cancels the bosonic ground state
energy $E_{0,0}$ in (12).

Generalization to multiple chiral multiplets is straightforward.

### Softly broken supersymmetry:

A simple way to explicitly break supersymmetry would be to add a small bosonic
potential, without any fermionic counterpart. Perhaps the simplest choice for
the modified potential would be

$\displaystyle V_{B}$ $\displaystyle=\left|\frac{\partial
W}{\partial\phi}\right|^{2}+\epsilon|\phi|^{2}\,,$ (16)

with
$\epsilon^{2}\ll\frac{\partial^{2}W}{\partial\phi^{2}}\Big{|}_{\phi=\phi_{0}}$,
for all extrema $\phi_{0}$ of the superpotential $W$. For multiple chiral
fields, $\epsilon^{2}$ should be much smaller than the smallest eigenvalue of
the Hessian $\partial_{i}\partial_{j}W$.

Now we determine the change in low energy dynamics due to $\epsilon|\phi|^{2}$
piece. The change is two fold, first the change in location of the potential
minima (which would generically have non-zero potential) and change in the
frequencies of the harmonic oscillators. Given an extremum $\phi_{0}$ of $W$,
let the shifted minima of $V_{B}$ be at $\phi_{0}+\varphi_{0}$. Expanding
$\phi=\phi_{0}+\varphi_{0}+\rho$, around $\phi=\phi_{0}$ we have

$\displaystyle{}V_{B}$
$\displaystyle\simeq|M|^{2}|\varphi_{0}+\rho|^{2}\,+\,\epsilon|\phi_{0}+\varphi_{0}+\rho|^{2}$
$\displaystyle=|M|^{2}|\varphi_{0}|^{2}\,+\,\epsilon|\phi_{0}+\varphi_{0}|^{2}\,+\,\left(\epsilon\phi_{0}^{*}+(|M|^{2}+\epsilon)\varphi_{0}^{*}\right)\rho\,+\,\left(\epsilon\phi_{0}+(|M|^{2}+\epsilon)\varphi_{0}\right)\rho^{*}+(|M|^{2}+\epsilon)|\rho|^{2}\,.$
(17)

Demanding $\rho=0$ to be a minima of $V_{B}$, we find
$\varphi_{0}=-\epsilon\frac{\phi_{0}}{|M|^{2}+\epsilon}$. Note the value of
the potential at the minima is now

$\displaystyle V_{0}$
$\displaystyle=|M|^{2}|\varphi_{0}|^{2}\,+\,\epsilon|\phi_{0}+\varphi_{0}|^{2}=\epsilon\frac{|\phi_{0}|^{2}}{|M|^{4}}+\dots\,.$
(18)

Since different minima will have different $\phi_{0},M$, they are non-
degenerate. Perhaps a better description would be near-degenerate as the
difference of energy among various minima is of $\mathcal{O}(\epsilon)$, which
we have taken to be very small.

Even the quantum ground state energies no more vanish and differ for different
minima. The bosonic oscillators now have frequencies
$\approx|M|+\frac{\epsilon}{2|M|}$. For the fermionic oscillators,
$\frac{\partial^{2}W}{\partial\phi^{2}}\Big{|}_{\phi=\phi_{0}}$ will be
replaced by

$\displaystyle\frac{\partial^{2}W}{\partial\phi^{2}}\Big{|}_{\phi=\phi_{0}+\varphi_{0}}$
$\displaystyle=M+\varphi_{0}\frac{\partial^{3}W}{\partial\phi^{3}}\Big{|}_{\phi=\phi_{0}}+\dots\,,$
(19)

which depends on the detail of $W$.

Anyhow, the bosonic and fermionic frequencies are different, thus the the
ground state energy is no more zero but some $\mathcal{O}(\hbar\epsilon)$
number. We have reinstated $\hbar$ to stress that this quantum ground state
energy is on top of a classical ground state energy $V_{0}$ (18).

Since the modified ground state energies depend on the detail of the
particular extrema of the superpotential, ground states corresponding to
different minima would be different. But the difference will be of
$\mathcal{O}(\epsilon)$, hence for small $\epsilon$, such difference will be
small as well, leading to large number of closely spaced low lying states, one
corresponding to each minimum of the superpotential. Taking perturbative
corrections as well as tunnelling effects, will change the detail. But the
broad picture, i.e. large number of densely packed low lying states will hold
true.

Note that excited states are separated by a gap of $\mathcal{O}(\hbar|M|)$, we
would like $V_{0}$ to lie well below this gap scale. While it is somewhat odd
to demand a classical potential minima to be small by quantum standards, there
is nothing inconsistent at a formal level. In an actual physical problem, the
supersymmetry breaking term itself might have quantum origins.

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11institutetext: Communication System Group, Department of Informatics,
University of Zurich22institutetext: Blockchain and Distributed Ledger
Technologies Group,
Department of Informatics, University of Zurich 33institutetext: Department of
Banking and Finance, University of Zurich

# Empirical and Theoretical Analysis of
Liquid Staking Protocols

Krzysztof Gogol 11 Benjamin Kraner 22 Malte Schlosser 33 Tao Yan 22
Claudio Tessone 22 Burkhard Stiller 11

###### Abstract

Liquid staking has become the largest category of decentralized finance
protocols in terms of total value locked. However, few studies exist on its
implementation designs or underlying risks. The liquid staking protocols allow
for earning staking rewards without the disadvantage of locking the capital at
the validators. Yet, they are seen by some as a threat to the Proof-of-Stake
blockchain security.

This paper is the first work that classifies liquid staking implementations.
It analyzes the historical performance of major liquid staking tokens in
comparison to the traditional staking for the largest Proof-of-Stake
blockchains. Furthermore, the research investigates the impact of
centralization, maximum extractable value and the migration of Ethereum from
Proof-of-Work to Proof-of-Stake on the tokens’ performance. Examining the
tracking error of the liquid stacking providers to the staking rewards shows
that they are persistent and cannot be explained by macro-variables of the
currency, such as the variance or return.

###### Keywords:

Liquid Staking DeFi Liquid Staking Risk Ethereum

##  1 Introduction

Table 1: Top 10 Crypto Assets by Staking Marketcap[26], [6] BC |  Token |  Lockup | Reward | Adj. Reward | MCap | LSPs
---|---|---|---|---|---|---
Ethereum[33] |  ETH |  $\infty$ | 4.82% | 5.04% | $35.8bn | 21
Cardano[11] |  ADA |  0 days | 3.25% | 0.14% | $9.2bn | 0
Solana[8] |  SOL |  1-3 days | 6.51% | -1.03% | $8.1bn | 9
BNB Chain[4] |  BNB |  90 days | 2.69% | 8.27% | $7.6bn | 8
Avalanche[10] |  AVAX |  14 days | 7.94% | 2.23% | $4.6bn | 5
Polygon[19] |  MATIC |  3-4 days | 6.24% | 3.37% | $4.0bn | 4
Polkadot[43] |  DOT |  28 days | 14.35% | 6.67% | $3.8bn | 1
Tron[29] |  TRX |  3 days | 3.78% | 1.7% | $2.6bn | 2
Cosmos Hub[35] |  ATOM |  21 days | 22.78% | 4.2% | $2.5bn | 1
Internet Computer[16] |  ICP |  180 days | 7.4% | -2.39% | $1.8bn | 0

Ethereum [33] successfully migrated from Proof-of-Work (PoW) to Proof-of-Stake
(PoS) consensus mechanism in 2022. After the Shanghai upgrade in 2023, it
became possible to unstake Ether (ETH) [13]. Within the Decentralized Finance
(DeFi) space, staking is seen as a relatively safe token allocation that
generates steady returns. The inherent disadvantage of the process is the
necessity to lock tokens for days or months, depending on the blockchain
(BC)1. _Liquid Staking Protocols_ (LSPs) manage to overcome this disadvantage
by constructing synthetic (pegged) tokens - _Liquid Staking Tokens_ (LSTs)
that accumulate rewards from the staking process without the need to lock any
assets. Lido [2], the first LSP, became the largest DeFi protocol in terms of
total value locked (TVL)[6]. Liquid staking has become the largest DeFi
category with 78 protocols and more than 16 billion USD TVL[7]. LSPs vary in
terms of implementation design, level of decentralization, governance, and the
validators’ selection process. MEV (Maximal Extractable Value)[34][38] attacks
are performed by validators who reorder transactions in the block in order to
generate profit. Some LSPs include the earning from MEV attacks in the LSTs
rewards.

The benefits of LSPs surpass just unlocking the staked assets. By making the
tokens liquid, LSPs allow to fully benefit from DeFi composability (”DeFi
Lego”) - compounding the returns from various DeFi protocols. LSTs can be
additionally allocated to other DeFi protocols. The typical LST allocation
strategy involves the liquidity pools in DEXs (Uniswap[31], Balancer[36]) or
interest rate protocols (Aave[1], Compound[40]).

LSTs belong to the broader class of DeFi protocols of synthetic (pegged)
tokens. DeFi synthetic tokens are digital assets that maintain the peg to the
target value. The best-known synthetic tokens are stablecoins. Stablecoins
maintain the peg to the fiat currency, typically one US Dollar, e.g.,
MakerDAO[42], Tether [28]. Other types of synthetic assets, wrapped tokens,
provide interoperability between blockchains. Wrapped Bitcoin[30] or Wrapped
Solana[8] allows exposure to native tokens of Bitcoin[37] and Solana [8],
respectively, on the Ethereum blockchain. The major risk related to synthetic
assets is the _de-peg risk_ \- the risk that a synthetic token loses the peg
to the target value. The depegs might be temporary, e.g., USDC[12] or DAI[42],
or permanent, UST[27], leading to the failure of the protocols. The collapse
of UST, the algorithmic stablecoin in 2022, resulted in the collapse of its
underlying blockchain and the entire ecosystem. Liquid staking tokens are
exposed not only to the slashing risk, related to staking, but also, as
synthetic assets, to the depeg risk.

#### Methodology and Contribution

Whereas liquid staking becomes the largest in terms of TVL DeFi category [7],
there is very little attention is paid to the matter in academic research. The
first and only paper [41], examines mainly the stability of the Lido peg on
the Ethereum and Solana blockchains. This paper makes the following
contributions.

  * •

It provided the taxonomy of the staking reward distribution currently employed
by the LSPs, resulting in three categories: rebase, reward, and dual model.

  * •

It analyzed data on market values and reserves of major LSTs that represent
90% of capital in liquid staking, and presented the first comparative study of
the historical returns of LSTs against native staking from the three largest
PoS blockchains - Ethereum, Solana, and Binance Smart Chain, finding that
market values of LSTs are affected by market events, such as Terra/Luna crash
or FTX insolvency.

  * •

It showed, by examining the liquid stacking providers to the staking rewards,
that tracking errors are persistent and cannot be explained by macro-variables
of the currency, such as the variance or return.

##  2 Background

This section introduces the key concepts behind blockchain technology, e.g.,
consensus mechanism, Proof-of-Stake (PoS) consensus mechanism and staking. It
further defines synthetic tokens, a class of DeFi protocols to which LSPs
belong, and explains the token price mechanisms and the difference between the
market and the peg values.

#### Blockchain and Proof-of-Stake

A blockchain is a computer network that maintains a distributed database. This
network only allows the appending of new transactions and forbids the deleting
or updating of any transactions. Transactions are grouped into blocks, and the
order of blocks in the database is maintained and verifiable using
cryptographic hash functions and digital signatures. For this paper, we define
a blockchain as a public, trustless, and permissionless distributed ledger. A
_consensus mechanism_ defines the rules for what constitutes a legitimate
transaction and a block. The dominant consensus mechanisms are PoW or PoS. The
consensus mechanism, based on game theory, economically incentives network
participants - miners in PoW and validators in PoS blockchains - to promote
network security. In PoW BCs, miners, to append a new block to the ledger, are
required to solve the cryptographic puzzle in a process that requires
electricity[37]. In PoS BCs, validators, to append a new block, must pledge
(”stake”) a certain amount of native tokens that act as collateral. In
exchange for adding a new block, validators receive a reward. In case of
malicious behavior, decreased up-time or latency validators are punished in a
process called _slashing_ by a fee from the staked tokens. _Staking_ is the
process of providing native BC tokens to validaters to participate in profits
from the block rewards. By design, the PoS blockchain requires validators and
_stakers_ to freeze their tokens for a period of time that may vary from days
to years. _Gas fee_ is paid to miners and validators for appending the
transaction to the BC.

_Ethereum Migration to PoS:_ On September 15, 2022 Ethereum completed its
transition from PoW to PoS consensus mechanism, also known ”the Merge”
upgrade. The process started in December 2020, with the launch of staking and
the Beacon Chain, and finished in April 2023, with the ”Shanghai” upgrade. As
of ”Shanghai” upgrade, it is possible to unstake the Ether [13].

_PoS Blockchains:_ The comparison between the top 10 PoS BCs in market
capitalization is presented in table 1. The table also presents the staking
rewards and lockup periods, among others. Until the ”Shanghai” upgrade in
April 2023, it was not possible to unstake the Ether. After the upgrade, the
unstaking process lasts ca 1-2 days, similarly to Solana chain. Currently,
there are no liquid staking protocols in Cardano, the second largest BC, as it
offers the automatic staking of all tokens held by a BC address without any
lock-up period.

Table 2: Synthetic tokens and their target (peg) values Synthetic Token |  Target
---|---
Stablecoins |  Fiat Currency, Commodity
Wrapped Tokens |  Cryptocurrency
Liquid Staking Tokens |  Staked Token

#### Synthetic (Pegged) Tokens

Synthetic (pegged) tokens in DeFi have their value pegged to the target
values, typically fiat currencies, commodities, or other cryptocurrencies
[39]. They track the price fluctuations of the target value[39], [32] and
their most well-known examples include stablecoins and wrapped tokens.
Stablecoins are pegged to fiat currency, typically 1 USD. Wrapped tokens
provide blockchain interoperability by having a peg to tokens minted at other
chains; e.g., WrappedSolana on the Ethereum blockchain is a synthetic token
with a peg to SOL - the native token Solana blockchain. LSTs are synthetic
tokens that peg the value to the staked native tokens and their accumulated
staking rewards.

_Fair and Market Value—_ There are various algorithms that synthetic tokens
use to maintain the peg to the target value. The most common is holding the
reserves of the target assets. This approach ensures that each synthetic token
has a potential redemption value, also known as the _peg value_. The fair
value is calculated by dividing the value of the reserves by the number of
tokens in circulation. However, tokens cannot always be freely exchanged at
this peg value. For example, stETH from Lido and cbETH from Coinbase have only
introduced this possibility since the Shanghai upgrade in April 2023, which
makes unstaking in the BC possible. The _market value_ of synthetic tokens is
the value at which tokens are traded on decentralized exchanges (DEX) or
centralized exchanges (CEX). The difference in value between the fair and
market value of synthetic tokens may be caused by market inefficiencies and
may result in arbitrage opportunities.

##  3 Liquid Staking Protocols

_Liquid staking_ is a tokenized representation of staked assets in PoS
blockchains, and consequently, its value is pegged to the staked tokens [41].
Liquid staking tokens (LSTs) are also referred to as _Liquid Staking
Derivatives_ (LSDs), whereas the protocol that mint LSTs are _Liquid Staking
Protocols_ (LSPs). Staking is the process of locking tokens in validators to
support the PoS blockchain security and, in exchange, earn profits [41].
Validators, network participants on the PoS blockchain, put those tokens at
stake when appending or attesting new transaction blocks to the blockchain.
The validator receives a reward for each correctly appended block, which is
redistributed between token providers (_stakers_). Dishonest behavior is
punished by a reduction in staked tokens (_slashing_). By design, the staked
tokens are locked for a specific period, making it impossible to use them in
other applications, reducing the token liquidity. By tracking the value of
staked assets, liquid staking tokens allow participating in the staking
benefits. Unlike staked tokens, liquid staking tokens are not locked but are
freely traded and can be used in DeFi applications.

Algorithm 1 Minting LST by LSP

1:Collect native tokens from the Depositor.

2:Stake deposited native tokens in the LSP’s validators.

3:Mint new LSTs, with the value of native tokens deposited, and provide LSTs
to the depositor.

Algorithm 2 Burning LST by LSP

1:Collect LSTs from the Depositor that redeems LSTs for the native tokens.

2:Unstake native tokens with the value of collected LSTs and provide unstaked
native tokens to the Depositor.

3:Burn the collected LSTs.

Algorithm 3 Arbitrage Strategy when market value $<$ fair value

1:Buy undervalued LSTs at DEX for the market value.

2:Burn LSTs at LSP for the peg value.

Algorithm 4 Arbitrage Strategy when market value $>$ peg value

1:Mint LSTs at LSP for the peg value.

2:Sell LSTs at the DEX for the market value.

Blockchain users have two possibilities to acquire LST: (i)buy LSTs at DEX or
CEX for the market price or (ii)mint LSTs directly at the LSP for a fair
price. LST minting and burning processes are presented by algorithms 1 and 2,
respectively. The blockchain user that mints the LSTs is called _Depositor_ ,
as he deposits the native tokens in the LSP in exchange for the LSTs. The
value of the newly minted LSTs is equal to the value of the deposited tokens,
so the fair value remains unchanged. LSP distributes the native tokens
deposited to the validators for staking. There are three models of
distributing the staking rewards to the holders of LSTs: (i) rebase, (ii)
reward, and (iii) dual model - these models are further described in the next
section. Similarly to acquiring LSTs, the blockchain user has two options to
redeem the LSTs: (i) sell LSTs at DEX or CEX for the market price or (ii) burn
LSTs directly at the LSP for a fair price. When the LSTs are redeemed at the
LSPs, the equivalent value of native tokens is unstaked and LSTs are burned.

_Arbitrage Strategies—_ Market inefficiencies could lead to a difference
between the fair and market value of synthetic tokens. Algorithms 4 and 3
present the arbitrage strategies that equal the fair and market values of the
LSTs. When the fair value of LSTs is higher than the market price,
arbitrageurs buy the LSTs at DEX for the market value and burn at LSP for the
fair value. In the opposite circumstance, the arbitrageurs mint the LSTs at
LSP for the fair value and sell at DEX for the market value, which is higher
than the fair value.

_Risks—_ The _de-peg risk_ , the major risk related to synthetic tokens,
including LSTs, is the risk of the token losing the peg to the target value.
Depeg might be caused by vulnerabilities in the technical or economical design
of the smart contract for synthetic tokens and can be exploited in hacking
attacks or oracle price attacks. In addition to the de-peg risk, LSTs are
vulnerable to the slashing risks caused by the LSP validators.

##  4 Taxonomy of Liquid Staking Protocols

This section presents the taxonomy of LSP. The taxonomy is straightforward, as
all information can be directly retrieved from the whitepaper or the source
code of the LSPs.

Figure 1: Taxonomy of Liquid Staking Protocol

###  4.1 Taxonomy by Token Model

The major differentiation between LSPs is the mechanisms for distributing the
staking rewards. LSPs apply three standard models for this purpose [3]:
(i)rebasing tokens, (ii)reward-bearing tokens, and (iii)3 dual token model.

_Rebase Tokens—_ In this model, LSP tokens keep the 1:1 connection to staked
tokens and inflate the supply to reflect the profit from stake. The most
popular rebase LSP is stETH, minted by Lido Protocol[2]. Lido every day
increases the balance of all stETH holders to reflect the additional ETH
earned as staking rewards. The advantage of rebasing LSP is its simple
functionality. The disadvantage is the lack of compatibility with other DeFi
protocols, e.g., DEXs, or interest rate protocols.

_Reward-bearing Token—_ In this model, the token increases its value to
reflect the staking yields and does not maintain a 1:1 peg to the native
token. Consequently, the reward-bearing token grows in value related to the
underlying native token. Reward-bearing tokens provide full compatibility with
DeFi protocols, and consequently, most LSPs follow this design:
RocketPool[21], Marinade [18], CoinBase Wrapped Staking [5].

_Dual Token—_ In the dual token model, the asset token is split from the stake
revenue it generates. For example, StakeWise [25] operates in the two-token
model: sETH2 - with a 1:1 peg to ETH - and rETH2 - which reflects the earned
staking rewards. The disadvantage of this model is the fragmented liquidity
between two tokens.

###  4.2 Taxonomy by Validator Selection

LSP also varies in the validator selection process: (i)whitelisting reputable
validators, (ii)credential-based, (iii)collateral. In the whitelisting model,
LSP adds only trusted validators to the pool of validators, e.g., Lido[2]. In
the credential-based model, the pool of validators is permissionless after
fulfilling the minimum criteria. In the last model, LSP requires validators to
post collateral to guarantee its performance, e.g., Rocket Pool[21]. The
collateral model hampers the growth of the validator pools by increasing the
minimum threshold for staking.

###  4.3 Taxonomy by Network Architecture

LSPs can operate in a single chain or in multiple chains. Single-chain
protocols are deployed and operate within one BC, e.g., . Rocket Pool[21] on
Ethereum. App-chain (or Solo-chain) is LSP that developed sovereign BC to
optimize its services, e.g., pStake[20]. Cross-chain protocols operate on
multiple BCs and allow the value to cross between BCs, e.g., Lido allows one
to purchase stETH on the Polygon BC.

###  4.4 Revenue Models and Token Utility

The most common revenue model for LSP is commission on staking yields with the
standard fee is 10%. The token utility includes (i)governance (ii)revenue
sharing (iii)validator collateral. _Governance Utylity-_ Holder of the LSP
utility tokens participates in the DAO votes. _Revenue Sharing-_ LSPs charge a
commission for their services and collect it in the protocol treasury. LSP
utility token holders can participate in the distribution of part of the
revenues from the treasury. _Validator Collateral-_ LSPs might require to
deposit collateral in their utility tokens. RocketPool[21] enabled
permissionless validation selection by requiring validators to post their
utility tokens - RPL tokens - as collateral. The deposited collateral is also
used as protection against slashing.

##  5 Data

The analysis is performed within the three largest PoS blockchains regarding
market cap: Ethereum[33], Solana[8], and BNB Chain[4]. First, we analyze the
performance of the LSTs in comparison with the staking of the corresponding
native tokens ETH, SOL and BNB. We calculated the daily performance of LSTs
based on the market price we sourced directly from UniSwap[31] for LSPs on
Ethereum and from Yahoo Finance for others. The reason we do not use the peg
price is that we model the realistic token allocation strategy that might be
performed by the DeFi user.

Figure 2: Comparison of selected LSTs with staking ETH [ETH]

Figure 3: Comparison of selected LSTs with staking BNB [BNB]

Figure 4: Comparison of selected LSTs with staking SOL [SOL]

Figure 5: Cumulative distribution function of excess returns (in percent) of
LSP tokens in Ethereum, Solana and BNB Chain.

###  5.1 Ethereum

The analysis includes reward-bearing tokens in the Ethereum chains. This
approach allows a comparison of how effectively the market price of LSTs
accumulates the ETH staking rewards. The value of re-base LSTs, such as stETH
or frxETH, tracks the value of 1 ETH and the staking rewards are distributed
daily to the token holder as additional tokens. Thus, tracking their value
does not allow to compare the accumulated rewards from staking. LSPs, such as
Lido or Frax, issue two LSTs: re-based and reward tokens, stETH and wstETH and
frxETH and sFRX, respectively. The reward tokens analyzed include wstETH Lido,
rETH Rocket Pool[21], cbETH Coinbase[5] and sfrxETH Frax[14]. Figure 4
presents the compounded daily returns of the reward-bearing LSTs on the
Ethereum blockchain against staking Ethereum.

_Dispersion Analysis—_ The dispersion analysis of ETH LSTs is presented in
figure 5. The figure presents the distribution of the excess return,
calculated as the absolute difference between the daily LST returns and the
daily returns from ETH staking. The fees charged by LSPs lead to a negative
shift in the distribution, the inclusion of MEV rewards in a rightward shift.
Eventually, the LSP returns are slightly negative, as indicated by a negative
median return. On a daily return basis, cbETH has the lowest tracking error,
while the highest deviation occurs for ankrETH tokens. Table 4 confirms this,
since the standard deviation of the excess return is the lowest for cbETH and
all tokens have a negative median.

_Terra and FTX collapses—_ The crypto market was severely impacted in 2022 by
adverse market situations. The first was the crash of the algorithmic USDT
stablecoin that caused the collapse of Terra BC and its entire ecosystem of
protocols (7 May 2020). Following the collapse of Terra, the centralized
exchange (CEX) - FTX - became insolvent (2 November 2022), freezing the
digital assets of its customers. In the period between the collapse of Terra
and FTX, both the market values of the stETH and rETH tokens performed worse
than traditional staking. In the aftermath of the FTX collapse, the rETH token
performed better than stake, which can be attributed to the investor’s
reluctance to centralized protocols such as FTX. Rocket Pools, due to its
permissionless selection of validators, is considered more decentralized than
the Lido protocol, which whitelists its validators. However, to protect
against dishonest validators, Rocket Pool requires validators to deposit the
collateral. The drop in the market value of LSTs might be caused by (i)market
inefficiencies in pricing the token or (ii)drop in the value of its reserves
(e.g., as a result of slashing or hacking). When the LST’s market value
differs from the peg value, the market is inefficient, and arbitrage exists.

_Peg Analysis—_ Until the Shanghei upgrade, there was no possibility to
unstake ETH. Consequently, the market value of LSTs pegged to staked ETH might
deviate from their peg value. Figure 6 illustrates the market and peg values
for the rETH token of RocketPool. In the period between the Terra-Luna
collapse (7 May 2022) and FTX collapse (Nov 2, 2022), the rETH was underpriced
compared to its peg value. If unstaking ETH would be possible in this period,
arbitrageurs could equal the market and peg values of rETH. Following the FTX
crash and until January 2023, the rETH tokens were overpriced, and the
arbitrage strategy existed, algorithm 4. The rETH could be minted at the
Rocket Pool Protocols for the peg value and sold at the DEX (UniSwap) for the
market value, which was higher than the peg value. The over-pricing of LSTs
from Rocket Pools can be attributed to the reluctance to the centralized
protocol following the FTX collapse.

Figure 6: Pegged value vs market value for rETH

###  5.2 Solana

We analyze the following LSPs in Solana BC - stSOL from Lido[2], mSOL from
Marinade[18], jSOL from JPool[17] and scnSOL from Socean[22]. All tokens are
reward-bearing and their value accumulates daily staking rewards from staking
SOL - the native token of the Solana BC. All LSPs are deployed to the Solana
BC. Figure 4 presents the compounded daily returns of these tokens against SOL
stakes. All tokens perform similarly, with the temporary drop in value in
November 2022 related to the collapse of the FTX exchange (2. Nov) that had a
significant involvement in the Solana BC and its ecosystem. Following the
Terra collapse, market values of LSTs on Solana were not traded undervalued
like ETH-based tokens, which can be explained by the possibility to unstake
SOL within days, compared to the lockup of ETH in staking until the Shangai
upgrade. The dispersion analysis - figure 5 \- shows that all LSTs on Solana
track the SOL staking rewards.

###  5.3 BNB Chain

We analyze the performance of LSTs tracking the staking rewards of native
token BNB on the Binance Chain: Ankr[9], Stader[23], pStake[20]. All tokens
are reward-bearing and accumulate the staking rewards daily in the token
value. Figures 4 and 5 present compounded daily returns of these tokens
against BNB staking, and their dispersion from the BNB staking returns,
respectively. BNB Chains offers 30, 60, and 90 day staking periods, with
higher staking rewards for the longer period. With the LSTs on BNB it is
possible to benefit from the higher rates.

##  6 Empirical Analysis

###  6.1 Model

We use multivariate regressions with trend coefficients to analyze potential
reasons for the deviations of the daily return and the premium/discount. We
follow [Hayashi] as a basis for the econometric models.

A multivariate regression tests the relationship of several independent
variables to a dependent variable with an ordinary least square estimation
method. The assumption is that there is a linear relationship between the
dependent variable $y$ and the n independent variables $x_{1},\dots,x_{n}$:

$y=\beta_{0}+\beta_{1}\cdot x_{1}+\beta_{2}\cdot x_{2}+\dots+\beta_{n}\cdot
x_{n}+\varepsilon,$

where $\beta_{1},\dots,\beta_{n}$ are the estimation coefficients for
$x_{1},\dots,x_{n}$ and $\varepsilon$ is the error term. The parameters
$\beta_{1},\dots,\beta_{n}$ are estimated in such a way that the sum of the
Euclidean norm is minimised and thus the best possible linear fit is found.

We use this model to test whether different variables have an impact on the
excess return of the LST $(Xs_{LST})$, defined as the difference between the
return of the LST and the staked base currency and how big this impact is. We
are particularly interested in whether the daily price change, the market
capitalization, the volume, or the standard deviation can explain part of the
excess returns. The price change is used to test whether stronger revaluations
or devaluations of the base currency have an impact on the excess return,
which can be caused by positive or negative news about that currency. The
market capitalization is used to check if the size of the LSP has an influence
on the excess return, while the traded volume is used to analyze if the excess
return depends on active or passive days. The standard deviation is used to
understand the behavior of the excess return in turbulent market phases. This
results in the following model

$Xs_{c}=\alpha+\beta_{1}\cdot\Delta_{\text{daily}}^{c}+\beta_{2}\cdot\sigma_{\text{monthly}}^{c}+\beta_{3}\cdot\sigma_{\text{daily
change}}^{c}+\beta_{4}\cdot M\beta_{5}\cdot V,$

where $c$ is a LST on either Ethereum, Solana or BNB, $\Delta_{\text{daily}}$
the daily price percentage price change in USD, $\sigma_{\text{monthly}}$ the
standard deviation of the last 30 trading days, $\sigma_{\text{daily change}}$
the daily percentage change in $\sigma_{\text{monthly}}$, MCap represents the
market capitalization, $V$ the daily volume, and $Xs_{c}$ represent the excess
returns.
In addition, we use heteroskedasticity-consistent standard errors (HC3) and
check the variance inflation factor (VIF) to control for multicollinearity.

Besides the daily excess return, we want to understand why a premium/discount
occurs and whether the above variables can explain when it exists. We have
seen in Figures 4 to 4 that the premium/discount has persisted over a longer
period of time. Due to this autocorrelation, we cannot perform a simple
multiple linear regression and we have to combine it with an autoregressive
model to analyze the premium. To determine the lags of the trend component, we
analyze the partial auto-correlation function of the premium/discount time
series to find out how many previous days have an impact on today’s premium. A
p-th order autoregressive process AR(p) is defined as

$y_{t}=\alpha+\beta_{1}\cdot y_{t-1}+\dots+\beta_{p}\cdot
y_{t-p}+\varepsilon_{t},$

where $y_{t-p}$ is the p-lagged value of $y$ and $\varepsilon_{t}$ is white
noise. We find out that for all protocols the effect disappears after 6 days
at the latest. Thus, we include a time lag of up to 6 days, which leads to the
model:

$\displaystyle\text{Premium}_{t}=$
$\displaystyle\alpha+\beta_{1}\cdot\Delta_{\text{daily}}^{c}+\beta_{2}\cdot\sigma_{\text{monthly}}^{c}+\beta_{3}\cdot\sigma_{\text{daily
change}}^{c}+\beta_{4}\cdot\text{MCap}+\beta_{5}\cdot V$ $\displaystyle+$
$\displaystyle\beta_{6}\cdot\text{Premium}_{t-1}+\dots+\beta_{1}1\cdot\text{Premium}_{t-6},$

where Premium is the difference between the market price and staking rate of
the LSP and we add 6 lagged trend variables to account for the autocorrelation
in the data. By including the lagged variables, we are not only able to see
how much the premium/discount depends on the past days, but also to isolate
the effect of the other variables from the autocorrelation.

###  6.2 Results

Tables 5 to 7 in the appendix show the OLS regressions, with the left half
showing the regressions on excess returns and the right half on
premium/discount.
For daily returns, the BNB and ETH regressions show similiar results. With the
exception of BNBx’s market cap and volume, none of the factors are relevant at
a 5% level of significance and the alpha cannot be distinguished from 0. This
means that most LSPs in ETH and BNB track the staking rewards very well on a
daily return level and the smaller deviations are not caused by any of our
macro-variables. For BNBx we have a positive correlation between market cap
and excess return. This means that the excess return was greater when the
market cap of BNBx increased. However, if this has led to a larger turnover of
the token, this has offset some of the gains. The picture is similar for
Solana, except that the $\Delta_{daily}^{SOL}$ factor is significant for all
tokens except stSOl. This means that the price development of Solana has been
reflected with an increased effect in the price of LSTs. Thus, on days with a
positive performance of the Solana-USD price, higher excess returns were
earned with the LSTs, while negative price movements of Solana reduced the
excess return or led to losses compared to staking. However, even for Solana,
the alpha is indistinguishable from 0, which means that the LSTs can track the
staking rate very well on a daily basis.

Regression of the premium/discount of LSPs yields a different result. The
premium strongly depends on the previous values. We see that especially the
two previous days show significant positive values. Thus, the premium tends to
persist if there was already a premium on the previous day. Thus, a 1
percentage point higher premium of rETH had a positive impact of 0.491
percentage points on the following day, 0.266 on the second day after and
still 0.09 on the third day at a significance level of 1%. In addition wstETH,
sfrxETH and mSOL tend to pay a premium when the underlying currency has a
higher market cap and a negative correlation with the volume traded. Apart
from scnSOl, an alpha significantly different from 0 was found for all LSTs at
a significance level of 1%. These results imply that the premium persists and
is not correlated with the macro-variables of the currency. This is an
indication that the discrepancy could be traded away via arbitrage.

##  7 Discussion

LSPs became the largest DeFi category in 2023 in terms of TVL [7] and Lido,
the first LSP, became the largest in terms of TVL protocol. The new LSPs, such
as RocketPool aim to address the centralized nature of Lido. Lido whitelists
the selected validators in a DAO vote and centrally manages the rewards,
inflows, and withdrawals. The new LSPs, such as RocketPool, request that the
validators deposit collateral to guarantee the honest behaviors of the
validators and cover the potential slashing losses. RocketPool, followed by
other LSPs, includes in LST rewards from the MEV attacks performed by the
validators, but this research found no impact of MEV rewards on the LST’s
performance. MEV (Maximum Extractable Value) refers to the profit earned by
validators by re-ordering the transaction in the block.

Algorithm 5 Compounding strategy for LST

1:Mint new LSTs at LSP with native tokens as collateral.

2:Deposit newly minted LSTs at AMM DEX in the liquidity pool comprising of
LST.

3:Deposit LP-tokens from AMM DEX in the DEX staking mechanism.

One of the DeFi promises is its composability, which allows compounding
returns from various DeFi protocols. The example DeFi strategy that involves
LST, presented in algorithm 5, involves allocating LSTs to DEXs. After minting
(or purchasing) LSTs, the tokens are locked in the liquidity pool at AMM DEX.
The liquidy pool includes the tokens pair: LST and stablecoin. In the last
step, the LP-tokens are allocated to the DEX’s staking mechanisms. Other LSTs
strategies might involve interest rate protocols (Aave, Compound) to leverage
or hedge the LSTs position. DeFi composability allows for a magnifying of the
returns of liquid staking.

This work defines staking as an integral part of PoS BC. However, some DeFi
protocols offer ”to stake” of their governance tokens or LP-tokens (e.g.,
Curve). These tokens are locked for a period of time to accumulate returns
from the DeFi protocol fees.

##  8 Summary

Liquid Staking Tokens (LSTs) are synthetic (pegged) tokens that are tokenized
representation of staked assets and consequently track the value of staked
tokens and staking rewards. This paper introduces the LST taxonomy,
differentiating between the (i) rebase, (ii) reward, and (iii) dual token
implementations. Subsequently, the reward-bearing tokens were empirically
analyzed for the largest blockchains in market capitalization: Ethereum,
Binance Smart Chain, and Solana.

This research found that LSTs on Solana and BNB Chain blockchains delivered
comparable returns to the staking of the native tokens, SOL and BNB,
respectively. Ethereum, until the Shanghai upgrade, did not support unstaking
ETH, which led to fluctuations in the market prices of LSTs, especially
following the Terra collapse and FTX insolvency.

After the FTX insolvency, the rETH tokens from RocketPool protocol traded
overpriced, compared to their peg value, which can be attributed to a
reluctance to the centralized protocols. RocketPool is considered to be more
decentralized than Lido protocol.

Table 3: Overview of analyzed LSPs LSP |  LST |  Target |  BC |  Token Model
---|---|---|---|---
Lido[2] |  stETH |  staked ETH |  Ethereum |  rebase
Lido[2] |  wstETH |  staked ETH |  Ethereum |  reward
Rocket Pool[21] |  rETH |  staked ETH |  Ethereum |  reward
CoinBase WSP[5] |  cbETH |  staked ETH |  Ethereum |  reward
Frax[14] |  sfrxETH |  staked ETH |  Ethereum |  reward
GETH[15] |  gETH |  staked ETH |  Ethereum |  reward
StakeWise[25] |  sETH2, rETH2 |  staked ETH |  Ethereum |  dual
Lido[2] |  stSOL |  staked SOL |  Solana |  reward
Marinade[18] |  mSOL |  staked SOL |  Solana |  reward
Socean[22] |  scnSOL |  staked SOL |  Solana |  reward
JPool[17] |  jSOL |  staked SOL |  Solana |  reward
Ankr[9] |  ankrBNB |  staked BNB |  BNB Chain |  reward
Stader[23] |  BNBx |  staked BNB |  BNB Chain |  reward
pStake[20] |  stkBNB |  staked BNB |  BNB Chain |  reward
StaFi[24] |  rBNB |  staked BNB |  BNB Chain |  reward

## Acknowledgements

The Authors thank DIVA and other liquid staking protocols for helpful
conversations.

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| wstETH | rETH | cbETH | sfrxETH | ankrETH | stSOL | mSOL | scnSOL | jSOL | ankrBNB | BNBx | STKBNB
---|---|---|---|---|---|---|---|---|---|---|---|---
Count | 547 | 500 | 226 | 87 | 110 | 550 | 542 | 538 | 501 | 116 | 228 | 243
Mean | 0.00000 | 0.00003 | 0.00014 | 0.00026 | 0.00082 | 0.00007 | 0.00011 | 0.00483 | 0.00007 | 0.00016 | 0.00014 | 0.00000
Std. | 0.00590 | 0.00920 | 0.00481 | 0.01479 | 0.01000 | 0.01107 | 0.01382 | 0.14620 | 0.01103 | 0.00950 | 0.00561 | 0.00732
Min. | -0.03046 | -0.11865 | -0.02215 | -0.08002 | -0.03112 | -0.09609 | -0.07849 | -0.71031 | -0.06145 | -0.03821 | -0.03296 | -0.03169
25% | -0.00245 | -0.00214 | -0.00172 | -0.00127 | -0.00379 | -0.00277 | -0.00462 | -0.01316 | -0.00350 | -0.00374 | -0.00245 | -0.00367
50% | -0.00009 | -0.00013 | 0.00006 | -0.00002 | -0.00009 | 0.00011 | 0.00011 | -0.00016 | 0.00011 | 0.00009 | 0.00040 | -0.00002
75% | 0.00239 | 0.00213 | 0.00207 | 0.00159 | 0.00506 | 0.00275 | 0.00429 | 0.01196 | 0.00356 | 0.00480 | 0.00342 | 0.00413
Max. | 0.03162 | 0.12566 | 0.02045 | 0.07999 | 0.03485 | 0.08239 | 0.07127 | 3.23583 | 0.10835 | 0.02320 | 0.01650 | 0.03097
Table 4: Descriptive Statistics of Excess Returns in different Liquified
Staking Tokens

| wstETH | rETH | cbETH | sfrxETH | akrETH | wstETH | rETH | cbETH | sfrxETH | ankrETH
---|---|---|---|---|---|---|---|---|---|---
const | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | -0.022∗∗∗ | -0.006∗∗∗ | 0.003∗∗∗ | 0.002∗∗∗ | 0.061∗∗∗
| (0.000) | (0.000) | (0.000) | (0.001) | (0.001) | (0.000) | (0.000) | (0.000) | (0.001) | (0.001)
$\Delta^{ETH}_{daily}$ | -0.001 | -0.011∗ | 0.014 | 0.032 | -0.010 | 0.002 | -0.008 | 0.014 | -0.086 | -0.012
| (0.005) | (0.006) | (0.012) | (0.034) | (0.038) | (0.005) | (0.006) | (0.013) | (0.077) | (0.040)
$\sigma_{\text{daily change}}^{ETH}$ | -0.005 | -0.005 | -0.009∗ | -0.046 | 0.005 | 0.001 | 0.002 | -0.007 | -0.092 | 0.013
| (0.006) | (0.005) | (0.005) | (0.037) | (0.025) | (0.005) | (0.007) | (0.004) | (0.072) | (0.025)
market cap | 0.000 | -0.000 | 0.000 | 0.031 | 0.001 | 0.002∗∗∗ | 0.002 | 0.005 | 0.058∗∗∗ | 0.026
| (0.001) | (0.001) | (0.002) | (0.019) | (0.010) | (0.001) | (0.002) | (0.003) | (0.017) | (0.016)
$\sigma_{monthly}^{ETH}$ | 0.003 | 0.011 | -0.011 | -0.277 | -0.166 | -0.044∗ | 0.030 | -0.030 | -0.172 | -0.270
| (0.014) | (0.020) | (0.022) | (0.243) | (0.165) | (0.023) | (0.035) | (0.019) | (0.255) | (0.346)
shift1 |  |  |  |  |  | 0.491∗∗∗ | 0.257∗∗ | 0.911∗∗∗ | -0.191∗ | 0.729∗∗∗
|  |  |  |  |  | (0.096) | (0.122) | (0.114) | (0.110) | (0.058)
shift2 |  |  |  |  |  | 0.233∗∗∗ | 0.266∗∗∗ | -0.056 | -0.209∗∗∗ | 0.065
|  |  |  |  |  | (0.065) | (0.053) | (0.172) | (0.045) | (0.141)
shift3 |  |  |  |  |  | 0.020 | 0.090∗∗∗ | -0.049 | -0.012 | 0.086
|  |  |  |  |  | (0.055) | (0.032) | (0.128) | (0.045) | (0.086)
shift4 |  |  |  |  |  | 0.179∗∗∗ | 0.052∗ | 0.261∗∗∗ | -0.054 | 0.005
|  |  |  |  |  | (0.051) | (0.031) | (0.085) | (0.079) | (0.111)
shift5 |  |  |  |  |  | 0.017 | 0.166∗∗∗ | -0.134∗∗ | 0.001 | -0.012
|  |  |  |  |  | (0.076) | (0.060) | (0.067) | (0.039) | (0.096)
shift6 |  |  |  |  |  | -0.046 | 0.047 | 0.004 | 0.017 | -0.000
|  |  |  |  |  | (0.052) | (0.047) | (0.056) | (0.037) | (0.073)
volume | -0.000 | -0.000 | 0.000 | 0.010 | -0.001 | -0.002∗∗ | -0.002 | -0.000 | -0.004 | -0.000
| (0.001) | (0.001) | (0.001) | (0.006) | (0.002) | (0.001) | (0.002) | (0.001) | (0.003) | (0.004)
Observations | 547 | 500 | 226 | 87 | 110 | 542 | 495 | 221 | 82 | 105
$R^{2}$ | 0.005 | 0.003 | 0.026 | 0.058 | 0.007 | 0.876 | 0.682 | 0.946 | 0.320 | 0.873
Adjusted $R^{2}$ | -0.005 | -0.007 | 0.004 | -0.001 | -0.041 | 0.873 | 0.674 | 0.943 | 0.213 | 0.857
Note: |  ∗p$<$0.1; ∗∗p$<$0.05; ∗∗∗p$<$0.01
Table 5: OLS Regression results for LSPs on Ethereum

| stSOL | mSOL | scnSOL | jSOL | stSOL | mSOL | scnSOL | jSOL
---|---|---|---|---|---|---|---|---
const | 0.000 | 0.000 | 0.000 | 0.000 | 0.004∗∗∗ | -0.001∗∗∗ | -0.001 | -0.007∗∗∗
| (0.000) | (0.000) | (0.001) | (0.000) | (0.000) | (0.001) | (0.001) | (0.000)
$\Delta^{SOL}_{daily}$ | 0.010 | 0.025∗∗ | 0.072∗∗ | 0.024∗∗∗ | 0.016 | -0.006 | 0.027∗ | 0.005
| (0.009) | (0.010) | (0.032) | (0.008) | (0.010) | (0.018) | (0.015) | (0.006)
$\sigma_{\text{daily change}}^{SOL}$ | 0.022 | 0.010 | 0.034 | -0.000 | -0.018∗∗ | -0.010 | -0.015 | -0.002
| (0.018) | (0.012) | (0.027) | (0.009) | (0.008) | (0.020) | (0.017) | (0.009)
market cap | 0.001 | 0.000 | 0.001 | 0.000 | 0.003∗ | 0.003∗∗ | 0.000 | 0.001
| (0.001) | (0.001) | (0.002) | (0.001) | (0.002) | (0.001) | (0.003) | (0.001)
$\sigma_{monthly}^{SOL}$ | 0.039 | 0.025 | 0.032 | 0.008 | 0.019 | 0.024 | -0.007 | 0.018
| (0.033) | (0.024) | (0.034) | (0.010) | (0.020) | (0.023) | (0.050) | (0.019)
shift1 |  |  |  |  | 0.335∗∗∗ | 0.343∗∗∗ | 0.105 | 0.346∗∗∗
|  |  |  |  | (0.093) | (0.111) | (0.072) | (0.133)
shift2 |  |  |  |  | 0.057 | 0.137∗ | 0.099 | 0.156∗∗∗
|  |  |  |  | (0.041) | (0.076) | (0.061) | (0.059)
shift3 |  |  |  |  | -0.041 | -0.098 | 0.103∗∗ | 0.090∗∗
|  |  |  |  | (0.082) | (0.062) | (0.050) | (0.045)
shift4 |  |  |  |  | -0.027 | -0.048 | 0.043 | 0.083∗
|  |  |  |  | (0.035) | (0.056) | (0.050) | (0.047)
shift5 |  |  |  |  | 0.016 | 0.021 | 0.046 | -0.081
|  |  |  |  | (0.021) | (0.052) | (0.053) | (0.062)
shift6 |  |  |  |  | 0.026 | -0.027 | 0.027 | -0.040
|  |  |  |  | (0.034) | (0.079) | (0.046) | (0.047)
total volume | -0.001 | 0.000 | -0.000 | -0.000 | -0.003 | -0.003∗∗ | -0.002 | -0.002∗
| (0.001) | (0.001) | (0.002) | (0.001) | (0.002) | (0.001) | (0.002) | (0.001)
Observations | 550 | 542 | 371 | 501 | 545 | 537 | 366 | 496
$R^{2}$ | 0.018 | 0.016 | 0.034 | 0.018 | 0.243 | 0.206 | 0.080 | 0.259
Adjusted $R^{2}$ | 0.009 | 0.007 | 0.021 | 0.008 | 0.228 | 0.190 | 0.052 | 0.242
Note: |  ∗p$<$0.1; ∗∗p$<$0.05; ∗∗∗p$<$0.01
Table 6: OLS Regression results for LSPs on Solana

| ankrBNB | BNBx | STKBNB | ankrBNB | BNBx | STKBNB
---|---|---|---|---|---|---
const | 0.000 | 0.000 | 0.000 | 0.012∗∗∗ | 0.023∗∗∗ | -0.008∗∗∗
| (0.000) | (0.000) | (0.000) | (0.001) | (0.000) | (0.000)
$\Delta^{BNB}_{daily}$ | -0.027 | -0.000 | -0.005 | -0.011 | -0.008 | -0.013
| (0.024) | (0.014) | (0.013) | (0.017) | (0.011) | (0.013)
$\sigma_{\text{daily change}}^{BNB}$ | 0.015 | -0.001 | 0.003 | 0.018 | 0.001 | 0.000
| (0.021) | (0.005) | (0.006) | (0.013) | (0.004) | (0.004)
market cap | -0.001 | 0.006∗∗ | 0.002 | -0.012 | -0.002 | -0.003
| (0.004) | (0.003) | (0.005) | (0.009) | (0.004) | (0.004)
$\sigma_{monthly}^{BNB}$ | -0.012 | 0.004 | 0.005 | -0.002 | -0.035 | 0.003
| (0.044) | (0.017) | (0.029) | (0.093) | (0.027) | (0.045)
shift1 |  |  |  | 0.190∗∗∗ | 0.485∗∗∗ | 0.411∗∗∗
|  |  |  | (0.064) | (0.057) | (0.054)
shift2 |  |  |  | 0.207∗∗ | 0.084 | 0.104∗
|  |  |  | (0.100) | (0.064) | (0.061)
shift3 |  |  |  | -0.025 | 0.104∗ | 0.153∗
|  |  |  | (0.093) | (0.058) | (0.089)
shift4 |  |  |  | -0.049 | 0.031 | -0.011
|  |  |  | (0.145) | (0.073) | (0.045)
shift5 |  |  |  | 0.076 | -0.034 | 0.104
|  |  |  | (0.106) | (0.062) | (0.079)
shift6 |  |  |  | 0.015 | 0.128∗ | -0.195∗∗∗
|  |  |  | (0.130) | (0.069) | (0.066)
total volume | -0.000 | -0.001∗∗ | -0.000 | -0.001∗ | -0.001∗ | -0.000
| (0.001) | (0.000) | (0.001) | (0.001) | (0.000) | (0.001)
Observations | 116 | 228 | 243 | 111 | 223 | 238
$R^{2}$ | 0.014 | 0.020 | 0.002 | 0.182 | 0.564 | 0.337
Adjusted $R^{2}$ | -0.030 | -0.002 | -0.019 | 0.092 | 0.541 | 0.305
Note: |  ∗p$<$0.1; ∗∗p$<$0.05; ∗∗∗p$<$0.01
Table 7: OLS Regression for LSPs on Binance