Yuhao Ye

Akiyoshi Yamada

Yuto Kinoshita

Jinhua Wang

Pan Nie

Liangcai Xu

Huakun Zuo

Masashi Tokunaga

Neil Harrison

Ross D Mcdonald

Alexey V Suslov

Arzhang Ardavan

Moon-Sun Nam

David Leboeuf

Cyril Proust

Benoît Fauqué

Yuki Fuseya

Zengwei Zhu

Kamran Behnia

High-field immiscibility of electrons belonging to adjacent twinned bismuth crystals

Bulk bismuth has a complex Landau spectrum. The small effective masses and the large gfactors are anisotropic. Moreover, at a high magnetic field, when only the lowest Landau levels remain occupied, the chemical potential does not stay constant. An added complexity arises from the existence of twin boundaries, which, by producing extra anomalies, further complexify the interpretation of the data in the extreme quantum limit. Here, we present an extensive study of low-temperature angle-dependence of magnetoresistance up to 65 T together with measurements of Nernst effect, ultrasound, and magneto-optics in bismuth. We found that all observed anomalies can be explained in a single-particle picture of a sample consisting of two twinned crystals tilted by 108 • . We show that a quantitative agreement between theory and experiment can be achieved only if one assumes that the two adjacent twinned crystals keep their own chemical potentials at a high magnetic field, despite a shift between chemical potentials as large as 68 meV at 65 T. This implies the existence of an energy barrier between adjacent twinned crystals reminiscent of a Schottky barrier between a metal and a semiconductor. We argue that this barrier is built by accumulating charge carriers of opposite signs across a twin boundary.

I. INTRODUCTION

Bismuth occupies a distinct place in the history of solid state physics (For reviews, see [START_REF] Dresselhaus | Electronic properties of the group V semimetals[END_REF][START_REF] Edel'man | Properties of electrons in bismuth[END_REF][START_REF] Issi | Low temperature transport properties of the group V semimetals[END_REF][START_REF] Fuseya | Transport properties and diamagnetism of Dirac electrons in bismuth[END_REF]). Its tiny Fermi surface was the first to be experimentally mapped out thanks to the easily detectable quantum oscillations it generates [START_REF] Shoenberg | The magnetic properties of bismuth, III. further measurements on the de Haasvan Alphen effect[END_REF][START_REF] Smith | Effective g factor of electrons and holes in bismuth[END_REF][START_REF] Bhargava | de Haas-van Alphen and galvanomagnetic effect in Bi and Bi-Pb alloys[END_REF][START_REF] Liu | Electronic structure of the semimetals Bi and Sb[END_REF]. Recent attention focuses on finding topologically non-trivial states both for the surfaces and the hinges of bismuth crystals [START_REF] Fu | Topological insulators in three dimensions[END_REF][START_REF] Drozdov | One-dimensional topological edge states of bismuth bilayers[END_REF][START_REF] Benia | Surface band structure of Bi1-xSbx(111)[END_REF][START_REF] Aguilera | Electronic phase transitions of bismuth under strain from relativistic self-consistent GW calculations[END_REF][START_REF] Ito | Proving nontrivial topology of pure bismuth by quantum confinement[END_REF][START_REF] Murani | Ballistic edge states in bismuth nanowires revealed by SQUID interferometry[END_REF][START_REF] Fuseya | Analytical solutions for the surface states of Bi1-xSbx (0 ≤ x ≲ 0.1)[END_REF][START_REF] Schindler | Higher-order topology in bismuth[END_REF][START_REF] Nayak | Resolving the topological classification of bismuth with topological defects[END_REF][START_REF] Hsu | Topology on a new facet of bismuth[END_REF]. Recent transport studies on macroscopic crystals have revealed a non-trivial variation of the magnetoresistance [START_REF] Collaudin | Angle dependence of the orbital magnetoresistance in bismuth[END_REF] and the Seebeck effect [START_REF] Spathelf | Magneto-seebeck effect in bismuth[END_REF] with the amplitude and the orientation of the magnetic field. A robust boundary conductance emerges at quantizing magnetic field, which is yet to be understood [START_REF] Kang | Boundary conductance in macroscopic bis-muth crystals[END_REF]. At the atomic scale, a bismuth layer on a NbSe 2 substrate was found to be a novel platform for detecting nanometric Turing patterns [START_REF] Fuseya | Nanoscale turing patterns in a bismuth monolayer[END_REF].

The rhombohedral structure of crystalline bismuth at ambient pressure is the outcome of lowering the cubic symmetry by pulling the cube along its body diagonal axis. Since this can be achieved along each of the four body diagonals, twin formation is common in bismuth crystals [START_REF] Savenko | Twinning of bismuth single crystals bombarded by boron ions[END_REF][START_REF] Bashmakov | Evolution of twinboundary geometry in bismuth crystals with an increase of load[END_REF] and twin boundaries in bismuth crystals were directly observed decades ago with a scanning tunneling microscope [START_REF] Edelman | STM revealing of twin microlayers with quantized width on cleaved bismuth surface[END_REF][START_REF] Edel'man | Scanning tunnel microscopy and spectroscopy of an atomically clean bismuth surface[END_REF].

The uncontrolled presence of such twin boundaries arXiv:2310.06685v1 [cond-mat.mtrl-sci] 10 Oct 2023 misled researchers seeking the signatures of collective electronic effects in bismuth. In the extreme quantum limit, when all carriers are confined to their lower Landau levels, a variety of electronic instabilities are conceivable [START_REF] Halperin | Possible states for a three-dimensional electron gas in a strong magnetic field[END_REF][START_REF] Macdonald | Strong-magneticfield states of the pure electron plasma[END_REF]. Experiments seeking such instabilities [START_REF] Behnia | Signatures of electron fractionalization in ultraquantum bismuth[END_REF][START_REF] Li | Phase transitions of Dirac electrons in bismuth[END_REF] explored bismuth beyond its quantum limit, achievable by applying a magnetic field exceeding 9 T along the trigonal axis, and found anomalies which, at first sight, could not be explained in the single-particle picture. Taken at face value, they were attributed to instabilities caused by electronic interactions. However, these interpretations in the case of bismuth were rejected subsequently. The computed Landau spectrum [START_REF] Alicea | Bismuth in strong magnetic fields: Unconventional Zeeman coupling and correlation effects[END_REF][START_REF] Sharlai | Origin of the peaks in the Nernst coefficient of bismuth in strong magnetic fields[END_REF][START_REF] Zhu | Angleresolved landau spectrum of electrons and holes in bismuth[END_REF][START_REF] Zhu | Landau spectrum and twin boundaries of bismuth in the extreme quantum limit[END_REF] was found to be in agreement with the experimental data and the hysteresis attributed to a first-order phase transition [START_REF] Li | Phase transitions of Dirac electrons in bismuth[END_REF] was not reproduced by torque magnetometry [START_REF] Fauqué | Hall plateaus at magic angles in bismuth beyond the quantum limit[END_REF] or magnetostriction [START_REF] Kuchler | Thermodynamic evidence for valleydependent density of states in bulk bismuth[END_REF] measurements.

It was later found that the additional anomalies [START_REF] Behnia | Signatures of electron fractionalization in ultraquantum bismuth[END_REF] were caused by the presence of a secondary twinned crystal tilted by 108 • with respect to the main crystal [START_REF] Zhu | Landau spectrum and twin boundaries of bismuth in the extreme quantum limit[END_REF]. Thus, up to 28 T, the single-particle picture proved to be sufficient to explain the experimental data [START_REF] Zhu | Landau spectrum and twin boundaries of bismuth in the extreme quantum limit[END_REF]. The same theoretical frame was then employed to explain how the evacuation of one or two electron pockets by a 40 T magnetic field, applied along the binary or the bisectrix axes, generates a sudden drop in magnetoresistance [START_REF] Zhu | Emptying Dirac valleys in bismuth using high magnetic fields[END_REF] and an anomaly in magnetization [START_REF] Iwasa | Thermodynamic evidence of magnetic-field-induced complete valley polarization in bismuth[END_REF].

The success of the single-particle picture in bismuth pushed beyond the quantum limit is to be contrasted with the case of graphite, another low-carrier semi-metal, in which the quantum limit is accessible in a threedimensional context. A succession of field-induced phase transitions have been detected in graphite [START_REF] Yaguchi | A high-magnetic-fieldinduced density-wave state in graphite[END_REF][START_REF] Fauqué | Two phase transitions induced by a magnetic field in graphite[END_REF][START_REF] Wang | Critical point for Bose-Einstein condensation of excitons in graphite[END_REF][START_REF] Zhu | Graphite in 90 t: Evidence for strong-coupling excitonic pairing[END_REF] indicating the capacity of strong Coulomb interaction to generate collective states [START_REF] Halperin | Possible states for a three-dimensional electron gas in a strong magnetic field[END_REF][START_REF] Macdonald | Strong-magneticfield states of the pure electron plasma[END_REF] in this material. The two elemental semi-metals differ in several aspects. Screening is weaker in graphite because of its lower dielectric constant. Spin-orbit coupling is much stronger in bismuth. Finally, graphite is much more anisotropic than bismuth, and its electron and hole Fermi pockets' longer axis are aligned.

In the case of bismuth-antimony alloys, possible signatures of field-induced electronic instability were recently observed [START_REF] Kinoshita | Field-induced reentrant insulator state of a gap-closed topological insulator (Bi1-xSbx) in quantum-limit states[END_REF]. They are to be distinguished from highfield anomalies due to the presence of twin boundaries seen before [START_REF] Banerjee | Transport anomalies across the quantum limit in semimetallic Bi0.96Sb0.04[END_REF].

Here, we present an extensive study of the Shubnikovde Haas effect, Nernst effect, ultrasound, and magnetooptics in bismuth at low temperatures and a magnetic field as strong as 65 T. We map the angle dependence of the Landau spectra for the whole solid angle and find that all anomalies can be explained in the single-particle picture, invoking the presence of a secondary twinned crystal. This includes the anomaly observed around 38 T when the magnetic field is applied along the trigonal axis of the main crystal [START_REF] Fauqué | Electronic instability in bismuth far beyond the quantum limit[END_REF], whose origin has not been hitherto identified. Thus, up to 65 T, there is no signature of instability caused by electron-electron interaction in Magnetoresistance data in the three planes of rotation: Magnetoresistance up to 65 T at 1.6 K for a magnetic field rotating in the (a) binary-trigonal, (b) bisectrixtrigonal, and (c) binary-bisectrix planes. The quantum limit of holes, when the penultimate hole Landau level is evacuated, occurs at 9 T when the magnetic field is along the trigonal axis and shifts upward as the field is tilted towards the binary or bisectrix axis.

bismuth. However, the quantitative agreement between theory and experiment is obtained thanks to a crucial assumption: that the two adjacent twinned crystals each have their own chemical potential µ, set by the orientation of the magnetic field respective to their crystalline axes. We will show that without this assumption, there is no agreement between the computed and the measured Landau spectrum. This unavoidable assumption leads us to conclude that electrons residing in two adjacent crys- tals do not mix up, despite a 68 meV difference in their chemical potentials at 65 T. We argue that this hypothesis points to the existence of a new type of p-n junction at the twin boundary of this semi-metal.

II. EXPERIMENTAL

Bismuth single crystals commercially obtained used in this study had typical dimensions of 1×2×0.5mm 3 and a Room-temperature-to-Residual-resistivity-Ratio (RRR) of ρ(300 K)/ρ(4.2 K)) ∼ 30 -100. Magnetoresistance (MR) measurements in pulsed magnetic field were carried out at Wuhan National High Magnetic Field Center (WHMFC) and in National High Magnetic Field Laboratory-Pulsed Field Facility (NHMFL-PFF) in Los Alamos. The samples were rotated in three distinct planes: binary-trigonal, bisectrix-trigonal and binarybisectrix. For each set of measurements, the current was applied along the axis perpendicular to the rotating plane. Our high-field MR data complemented the Nernst data obtained below 28 T and reported previously [START_REF] Zhu | Angleresolved landau spectrum of electrons and holes in bismuth[END_REF][START_REF] Zhu | Landau spectrum and twin boundaries of bismuth in the extreme quantum limit[END_REF]. Additional angle-dependent Nernst measurements up to 45 T were also carried out in the hybrid magnet of the NHMFL DC Facility in Tallahassee. Ultrasound measurements have been collected at Laboratoire National des Champs Magnétique Intense (LNCMI-Toulouse) with pulsed magnetic field up to 54 T. Longitudinal ultrasonic waves were generated using commercial LiNbO 3 36 • Y-cut transducers glued on a polished surface. A standard pulse-echo technique was used to determine the change in the sound velocity and attenuation. Magneto-optical measurements involving polarized light (for more details, see reference [START_REF] Katakura | Development of high-speed polarizing imaging system for operation in high pulsed magnetic field[END_REF]) were conducted at the International MegaGauss Science Laboratory, the Institute for Solid State Physics at the University of Tokyo. The resulting image was captured using a high-speed camera. G r e n o b l e T a l l a h a s s e

S

x y ( a . u . ) [START_REF] Fauqué | Electronic instability in bismuth far beyond the quantum limit[END_REF]. The measurements presented in all panels were performed in the bisectrix-trigonal plane and zero degree corresponds to field orientation along the trigonal axis.

B ( T ) -3 1 °- 1 5 °0 °1 7 °2 9 °( a ) .

III. RESULTS

Angle-dependent magnetoresistance in three planes:

Fig. 1 shows magnetoresistance along three planes, namely binary-trigonal (bin.-tri.), bisectrixtrigonal (bis.-tri.), and binary-bisectrix (bin.-bis.) at 1.6 K. Magnetoresistance shows oscillations which correspond to evacuations of the Landau levels with magnetic field. The different orientations for these planes are indicated. When the magnetic field is along the trigonal axis, MR shows a minimum around 9 T, which corresponds to the evacuation of the penultimate Landau level of holelike carriers [START_REF] Bompadre | Unambiguous determination of the g factor for holes in bismuth at high B/T[END_REF]. As the magnetic field is tilted off the trigonal axis, the cross-section of the hole Fermi pocket perpendicular to the magnetic field increases and this minimum shifts to higher fields. In the binary-bisectrix of Fig. 1(c), the drops are evident around 40 T because of the emptying of Dirac pockets [START_REF] Zhu | Emptying Dirac valleys in bismuth using high magnetic fields[END_REF].

Fig. 2 presents the second derivative of our magnetoresistance next to the Nernst measurements (S xy ) for three high-symmetric planes [START_REF] Zhu | Landau spectrum and twin boundaries of bismuth in the extreme quantum limit[END_REF]. The two types of data are consistent with each other, but the sensitivity of the Nernst results is relatively higher. Each peak in the Nernst response or in the second derivative of magnetoresistance corresponds to an intersection between an electron (or a hole) Landau level and the chemical potential. Note that the data have been normalized to the largest amplitude of each data set. Fig. 2 (a), (b), and (c) show the data for the magnetic field rotating in the binary-trigonal, bisectrix-trigonal, and in the binarybisectrix planes, respectively. In the corner of the panel, the schematic diagram shows the measurement configurations for MR. The quantum oscillations pattern is relatively simple and easily understood when the field is along the trigonal axis [START_REF] Behnia | Oscillating Nernst-Ettingshausen effect in bismuth across the quantum limit[END_REF]. The hole Landau peaks can be easily identified as their amplitudes are the largest, except for a sudden jump around 40 T along the trigonal axis.

Consistency of measurements: We verified the consistency between diverse techniques when the field was rotated in the bisectrix-trigonal plane and the data obtained on different samples. Fig. 3(a ing similar peak positions. Specifically, there are ∼40 T anomalies in S xy , in d 2 ρ dB 2 , and in ∆v. The slight difference in positions at θ ≈ 15 • is consistent with the experimental margin in measuring the angle between the magnetic field and the sample in three different experiments. We also notice that the anomalies in sound velocity and ultrasound attenuation occur at slightly different magnetic fields [START_REF] Akiba | Quantitative evaluation of Dirac physics in PbTe[END_REF].

Comparison with theory: The angle-dependent Landau spectrum can be described by a theoretical model that considers electron pockets at the L points of the Brillouin zone in an extended Dirac Hamiltonian and the hole pocket at the T point as carriers with parabolic dispersion and highly anisotropic g-factor [START_REF] Zhu | Landau spectrum and twin boundaries of bismuth in the extreme quantum limit[END_REF][START_REF] Zhu | Emptying Dirac valleys in bismuth using high magnetic fields[END_REF][START_REF] Zhu | Nernst quantum oscillations in bulk semi-metals[END_REF][START_REF] Fuseya | Origin of the large anisotropic g factor of holes in bismuth[END_REF]. In order to explain additional and unexpected anomalies, one needs to assume the presence of an additional sec-ondary twinned domain [START_REF] Zhu | Landau spectrum and twin boundaries of bismuth in the extreme quantum limit[END_REF][START_REF] Zhu | Emptying Dirac valleys in bismuth using high magnetic fields[END_REF]. In the high field limit, when only a single Landau level is occupied, the interband coupling becomes important and plays a significant role in setting the details of the spectrum [START_REF] Zhu | Emptying Dirac valleys in bismuth using high magnetic fields[END_REF].

The twinned sample originates from the distortion by the symmetry-lowering operation due to Peierls transition [START_REF] Peierls | More Surprises in Theoretical Physics[END_REF]. Since the Schmid factor is 0.48 in bismuth [START_REF] Ostrikov | Effect of carbon ion bombardment and surface oxidation on the twinning rate of single-crystal bismuth[END_REF], applying an external load [START_REF] Bashmakov | Evolution of twinboundary geometry in bismuth crystals with an increase of load[END_REF] and thermal cycling [START_REF] Ostrikov | Twinning in bismuth single crystals induced by thermal cycling[END_REF] could relatively easily induce twins compared to other materials [START_REF] Ostrikov | Effect of carbon ion bombardment and surface oxidation on the twinning rate of single-crystal bismuth[END_REF]. Thus, the strain appearing during sample cleaving by a blaze along cleavage basal trigonal (0001) plane in liquid nitrogen could introduce twinned domains on the sample and these twinned domains should mainly be on the surface. This could be a reason that a smaller RRR was found in a sample with a smaller dimension [START_REF] Galev | Exponential temperature dependence of the coefficient of the Nernst-Ettingshausen transverse effect in bismuth[END_REF], since the ratio of the one plane area to the volume is smaller in a smaller sample. When the sample is twinned, besides the existing principal trigonal, a quasi-trigonal axis will be present.

Fig. 4 compares the theoretical prediction with experimental peaks for three high symmetrical planes. The experimental peaks depicted by open symbols are compared with the theoretical results, which are presented by lines from both the main and a secondary tilted crystal. For clarity, we have omitted the electron peaks below 28 T, which have been identified earlier [START_REF] Zhu | Landau spectrum and twin boundaries of bismuth in the extreme quantum limit[END_REF]. Beyond 28 T, we included all peaks associated with electrons from both domains. Fig. 4(a) sketches the main and twinned crystals with their Fermi surfaces. The tilt angle between the two crystals is 108 • [START_REF] Zhu | Landau spectrum and twin boundaries of bismuth in the extreme quantum limit[END_REF]. The common binary axis is also shown. All theoretical results of the hole and twinned electron Landau level 0 e-are included.

The theoretical model here is fundamentally the same as the one used before [START_REF] Zhu | Angleresolved landau spectrum of electrons and holes in bismuth[END_REF][START_REF] Zhu | Landau spectrum and twin boundaries of bismuth in the extreme quantum limit[END_REF][START_REF] Zhu | Emptying Dirac valleys in bismuth using high magnetic fields[END_REF], except that the tensor, which is the correction to the additional g-factor in the lowest Landau level V ′ , was adjusted a little to fit the experimental Landau spectra better. The V ′ 3 was adjusted to -0.0025 from -0.0625 [START_REF] Zhu | Emptying Dirac valleys in bismuth using high magnetic fields[END_REF]. As seen in Fig. 4, whereas a slight displacement occurs, which may be due to misalignment during measurement, there is a relatively good agreement between the theory and the experiment. Some more peaks may be due to the electron Landau levels, which have not been clearly identified in the figures, see Fig. 4(c). We may conclude, therefore, that the Landau spectrum of bismuth corresponds to what is expected in the single-particle theory. No collective effect due to electron-electron interaction is visible up to 65 T. Because of extreme sensitivity, nullification of the Zeeman splitting and of holes along the trigonal axis, the experimentally resolved hole peaks are split due to a small residual misalignment in the case of binary-bisectrix plane (see Fig. 4(d)). The twinned crystal shares one common binary axis with the original crystal, but tilts its trigonal axis by 108 • off [START_REF] Zhu | Emptying Dirac valleys in bismuth using high magnetic fields[END_REF]. This is demonstrated by Fig. 4(d) at zero degrees where two sets of hole peaks merge. The additional peaks vanish as the two crystals are in the same configuration regarding the magnetic field and possess an identical Landau spectrum.

The twinning scenario provides an explanation for the previously reported drop in the magnetoresistance for a magnetic field of 38 T along the trigonal axis of the main crystal [START_REF] Fauqué | Electronic instability in bismuth far beyond the quantum limit[END_REF]. We can identify it as the result of emptying the electron pockets of the twinned domain. As seen in Fig. 4(b) and Fig. 4(c), the magenta dashed line, which represents the intersection between the 0 e-Landau level of the twinned domain and the chemical potential happens to be at ∼ 40 T when the magnetic field is along the trigonal axis of the main crystal. Thus, there is no ultra-quantum feature in the main crystal beyond the quantum limit of holes, which occurs at 9 T along the trigonal axis.

Twins observed by magneto-optics: We confirmed the presence of twin boundaries in our bismuth samples with a magneto-optical probe and found that pulsed magnetic fields have no influence on the domain population. The in-situ images are shown in Fig. 5. The crystal was illuminated with polarized light and the image was recorded using a high-speed camera in pulsed magnetic fields up to 30.8 T oriented along the trigonal axis. The field profile is shown in Fig. 5(d). Images are taken at the magnetic field at the times represented by red circles. Three typical images taken at 0 T, 30.8 T, and 0 T after the pulse are shown in Fig. 5 (a), (b), and(c), respectively. The binary axis common to the two domains is labelled in Fig. 5(a). The twinned domains occupy a large area close to one-third of the surface.

Independent or common chemical potentials for main and twinned crystals: Let us now turn our attention to the most striking aspect of our results. Since the charge is conserved, any change in the density of electrons is to be compensated by an identical change in the density of holes. This leads to a shift in the position of the chemical potential with increasing magnetic field [START_REF] Smith | Effective g factor of electrons and holes in bismuth[END_REF][START_REF] Zhu | Angleresolved landau spectrum of electrons and holes in bismuth[END_REF][START_REF] Zhu | Landau spectrum and twin boundaries of bismuth in the extreme quantum limit[END_REF][START_REF] Zhu | Emptying Dirac valleys in bismuth using high magnetic fields[END_REF]. This field-induced shift strongly depends on the orientation of the applied magnetic field. The main crystal and its twinned domain are tilted by 108 degrees. Therefore, the magnitude of the field-induced This conclusion implies a discontinuity in chemical potential and in carrier density across the twin boundary, which is reminiscent of a Schottky barrier. Remarkably, when the magnetic field is along the trigonal axis, the difference in chemical potentials increases up to 68 meV at 65 T (see Fig. 6 (c)). This is remarkable given that the twin boundary width is as narrow as 1 nm [START_REF] Edel'man | Scanning tunnel microscopy and spectroscopy of an atomically clean bismuth surface[END_REF][START_REF] Edelman | Tunneling spectra at terrace boundaries on the bismuth surface[END_REF]. This raises the following question: What impedes electrons belonging to adjacent crystals from mixing up? We now propose a possible scenario explaining this effect. It is well known that the bands are bent near the interface between electronically different solids [START_REF] Kittel | Introduction to solid state physics[END_REF].

The typical examples are shown in Fig. 7 (a) for the p-n junction and (b) for the Schottky junction. The former is a junction between semiconductors, and the latter is a junction between a metal and a semiconductor. In each case, chemical potentials are different in each side when separated (upper panels of Fig. 7). When they are joined, the carriers are forced to move to make µ common. For example, holes move from p-type to ntype, and electrons move from n-type to p-type in the p-n junction. The n-type band is shifted upward and the p-type band downward by the potential gradient due to the carrier transfer across the interface, resulting in the band bending with a common µ as shown in Fig. 7 (a).

A similar band bending is also expected in the present twin boundary between Bi crystals. At zero field, µ is shared between the main and twinned crystals. Under a strong magnetic field, µ of the main crystal shifts differently from that of the twinned one, as shown in Fig. 6 (c). This situation can be expressed by the model where two electronically different semimetals are joined at the twin boundary as depicted in Fig. 7 (c).

Suppose that µ of the main crystal is lower than that of the twinned one when two semimetals are separated. When they are joined, the carriers are forced to move to make µ common. As a result, both the electron and hole bands are bent, as if two types of p-n junction coexist. By this band bending, the relative levels between µ, the conduction and the valence bands are kept in each crystal as before joining. The Landau level spectrum of this twinned p-n junction is equivalent to that of the case where the independent chemical potentials are employed. Eventually, the mystery of the apparent jump of µ is solved by this scenario of the twinned p-n junction.

For the twinned p-n junction scenario, we assumed the electric potential difference between the main and twinned crystals. One may question whether an inhomogeneous electric field can exist inside a (semi) metal. The standard answer would be "No" because finite current will screen the potential difference. However, Landauer argued against this simplistic view [START_REF] Landauer | Residual resistivity dipoles[END_REF][START_REF] Landauer | Spatial variation of currents and fields due to localized scatterers in metallic conduction[END_REF]. Even in a metal, a point-like scattering center inside a metal will create an unscreened dipolar field, i.e., the electric field is inhomogeneous near the point-like defects. In the present case, the twin boundary, a line or plane defect, will play a similar role to the point-like scattering center of Landauer's theory.

In this scenario, the Landau eigenvalues curve in the vicinity of a twin boundary in the same way as they bend near the edges of a 2D electron gas [START_REF] Macdonald | Hall voltage and current distributions in an ideal twodimensional system[END_REF]. Any time of such Landau level bending results in edge currents, the continuity equation causes the edge charge and currents to remain close to the edge but distributed in real space in a 2D case [START_REF] Macdonald | Hall voltage and current distributions in an ideal twodimensional system[END_REF]. In the present 3D case, the continuity equation strictly confines the currents and any associated charge variations to the vicinity of the twin boundary. The currents flow along the twin boundaries and essentially account for small differences in magnetization between twins, which should be present due to differences in Landau level filling factors related to the difference in chemical potential. The equilibrium distribution of currents and charge ultimately depends on a balance between the Lorentz force of the current and the twin boundary potential that causes the Landau level eigenstates to curve in real space. These currents flowing along the twin boundary result from a novel form of extended bound state associated with the twin junction. This line of inquiry deserves further investigations.

In summary, we have carried out angle-dependence of magnetoresistance along three high symmetric planes, and mapped out their Landau spectra up to 65 T. All anomalies including the one observed at ∼40 T for a magnetic field oriented along the trigonal axis could be explained by assuming the presence of a second twinned crystal and the two crystals keep separate chemical potentials. We suggested an interpretation of this result by a scenario in which a p-n junction builds up at the twin boundary-areas. This implies the existence of an inhomogeneous electric field in the vicinity of twin boundaries, which can be checked by future experiments.

n a r y 3 b i n a r y 2 b i n a r y 1 B ( T ) b i s e c t r i x 1 b i s e c t r i x 2 b i s e c t r i x 3 FIG. 1 .

 3211231 FIG.1. Magnetoresistance data in the three planes of rotation: Magnetoresistance up to 65 T at 1.6 K for a magnetic field rotating in the (a) binary-trigonal, (b) bisectrixtrigonal, and (c) binary-bisectrix planes. The quantum limit of holes, when the penultimate hole Landau level is evacuated, occurs at 9 T when the magnetic field is along the trigonal axis and shifts upward as the field is tilted towards the binary or bisectrix axis.

FIG. 2 .

 2 FIG.2. Experimental Landau spectrum for three rotating planes: (a)-(c). Nernst voltage (Sxy) vs magnetic field measured in a magnetic field up to 12 T (in a superconducting magnet), up to 28 T in resistive water-cooled magnet (curves in blue) along with the second derivative of the magnetoresistance measured in pulsed field experiments above 28 T up to 65 T (curves in red) for the three planes of rotation: binary-trigonal, bisectrix-trigonal, and binary-bisectrix planes, respectively. The data have been normalized to their highest amplitude for each data set. The 0 • and 90 • represent the field along trigonal and binary/bisectrix for the (a)/(b). The 0 • and 90 • represent the field along binary and bisectrix for the (c). The sketches show the direction of the orientation of the magnetic field and electrical/heat current for each plane of rotation. Note that in all the experiments, the electrical or the heat current is along the direction perpendicular to the plane of rotation.

FIG. 3 .

 3 FIG. 3. Consistency among different probes and samples: (a) Nernst voltage (Sxy) measured in Grenoble up to 28 T (dashed lines) compared with the Nernst voltage measured up to 45 T (solid lines) for several orientations of the magnetic field. (b) Nernst voltage (blue) compared with the second derivative of the magnetoresistance (red), and the sound velocity (black) vs. magnetic field at 0 • (c) Same for 15 • . Peaks in Sxy, and d 2 ρdB 2 are concomitant with a minimum in ∆v. The arrows indicate the ∼40 T anomaly observed previously[START_REF] Fauqué | Electronic instability in bismuth far beyond the quantum limit[END_REF]. The measurements presented in all panels were performed in the bisectrix-trigonal plane and zero degree corresponds to field orientation along the trigonal axis.

FIG. 4 .

 4 FIG. 4. The experimental and theoretical Landau spectrum: (a) The illustration of the twinned structure in bismuth. The two crystals tilt from each other by 108 • and share one common binary. (b)The Landau spectrum of the main and twinned samples from holes according to theory (solid lines) and experiment (symbols) for binary-trigonal, (c) bisectrix-trigonal, and (d) binary-bisectrix planes. The red and blue symbols are for the peaks from the holes of the main and twinned crystals from the Fig.2, respectively. The red and blue lines represent the calculated Landau levels from holes of the main and twinned domains, respectively. The magenta dashed lines are for calculated 0e-electron Landau level which corresponds to the large drop in magnetoresistance as the field is along trigonal. The other electron peaks below 28 T are omitted for clarity, see the ref[START_REF] Zhu | Landau spectrum and twin boundaries of bismuth in the extreme quantum limit[END_REF].

FIG. 5 .

 5 FIG. 5. No detectable change of twins under pulsed magnetic field: (a), (b), and (c) show photos at the specific time at 0 T, 30.8 T, and 0 T after the pulse, indicated in the (d). The crystalline orientations are marked in the (a). (d). The time pattern of the magneto-optical measurements performed at 8.1 K. The magnetic field pulse for the imaging experiment lasts ∼ 5 ms. Images were taken every 0.2 ms as marked by red open circles.

FIG. 6 .

 6 FIG.6. Comparison between theoretical and experimental results for independent and common chemical potential: The Landau spectrum in the bisectrix-trigonal plane. In (a), µ in each crystal is independent of the other. In (b), the two crystals share a common µ. The calculated Landau spectrum and the experimental results agree in (a), but radically differ in (b). (c) The chemical potential shift for the main and twinned sample according to the theoretical results with independent µ. The difference becomes as large as 68 meV at 65 T.

FIG. 7 .

 7 FIG. 7. Schematic of band bending in p-n junctions, Schottky junction and newly proposed twinned p-n junction in semimetal bismuth before and after joining: Band bending in (a) the p-n junctions between two semiconductors; (b) the Schottky junction between a metal and a n-type semiconductor; (c) the twin boundary in semimetal bismuth

This work was supported by The National Key Research and Development Program of China (Grants No.2022YFA1403500 and 2016YFA0401704), the National Science Foundation of China (Grant No. 11574097 and No. 51861135104), and by directors funding grant number 20120772 at LANL. N.H. and RMcD acknowledge support from the US-DOE BES 'Science of 100T' program. B. F. acknowledges support from Jeunes Equipes de l'Institut de Physique du Collége de France (JEIP). K. B. was supported by the Agence Nationale de la Recherche (ANR-19-CE30-0014-04). Y. F. was supported by JSPS KAKENHI grants 23H04862, 23H00268, and 18KK0132. Work performed at National High Magnetic Field Laboratory was supported by National Science Foundation Cooperative Agreements DMR-0654118 and DMR-2128556 and the State of Florida. M. T. acknowledges support from the JSPS KAKENHI grant No. 23H04862. * zengwei.zhu@hust.edu.cn


Hansjörg Albrecher
email: hansjoerg.albrecher@unil.ch

Dina Finger
email: dina.finger@unil.ch

Pierre-O Goffard
email: goffard@unistra.fr

Empirical risk analysis of mining a Proof-of-Work blockchain

The process of mining blocks on a blockchain utilizing a Proof-of-Work consensus mechanism carries inherent risks, particularly when the operational expenses associated with mining exceed the rewards earned. Building on previous findings on mining in pools, this paper delves into the question of whether the theoretical formulas for the ruin probability and the expected value of future surplus obtained under particular model assumptions are indeed validated empirically. In particular, we include the presence of transactions fees in the block rewards in our analysis. We also provide algorithms to fit the involved generalized hyperexponential distributions to actual data. Moreover, we perform a sensitivity analysis for different factors of interest, and we quantify the relevance of incorporating temporal dependence and transaction fees in the model.

Introduction

A blockchain is a data ledger which is maintained by a Peer-to-Peer network. The database entries, referred to as transactions, are recorded by batches called blocks resulting from the application of a consensus protocol. In the case of the bitcoin, the consensus protocol is the Proof-of-Work. The network participants, called miners, compete to solve a cryptoproblem via a trial and error approach.

Computers are running 24/7 which consumes a lot of electricity. This operational cost is borne by the miners and compensated by a reward expressed in cryptocurrency units whenever a new block is found. The stability of blockchain systems relies heavily on this incentive mechanism. The balance between cost and reward is a stochastic process, which we model as

R t = u -C t + B t , t ≥ 0, (1)
where the initial capital u is augmented by the income (B t ) t≥0 net of the expenses (C t ) t≥0 . Such models were studied in [START_REF] Albrecher | On the profitability of selfish blockchain mining under consideration of ruin[END_REF][START_REF] Albrecher | Blockchain mining in pools: Analyzing the trade-off between profitability and ruin[END_REF][START_REF] Li | A mean field games model for cryptocurrency mining[END_REF] assuming that

C t = c • t, for c, t ≥ 0,
and

B t = Nt i=1 U i , t ≥ 0,
where (N t ) t≥0 is a Poisson process and the block rewards B i form a sequence of positive random variables. One goal of this paper is to propose a more accurate model for these rewards.

Block rewards consist of a protocol-specified bounty augmented by transactions fees. When passing a transaction, users will typically attach a transaction fee to it. This fee mostly depends on the transaction volume, since each block has a fixed allotted space inside. The pending transactions are stored in the memory pool (often abbreviated as mempool), where they await confirmation, hereby forming a queue. The transaction fee level closely relates to the network congestion that may be tracked by looking at the mempool size. Algorithms have been developed to inform the users of the appropriate transaction fee levels, see for instance the bok of Antonopoulos [START_REF] Antonopoulos | Mastering Bitcoin: unlocking digital cryptocurrencies[END_REF]Chapter 5]. Although common practice suggests that miners prioritize transactions with the highest transaction fee per byte rate, some deviations can be observed due to potential arbitrage opportunities linked to include specific transactions or ordering them in a given manner. Such considerations are beyond the scope of the present study, and we refer the reader to the work of Messias et al. [START_REF] Messias | Selfish & opaque transaction ordering in the Bitcoin blockchain: the case for chain neutrality[END_REF]. In the bitcoin blockchain, the impact of the transaction fees is currently still minor when compared to the bounty for finding a new block. However, as the reward gets halved approximately every four years, the need to understand the underlying dynamics of transaction fees will become pivotal in the future. In Carlsten et al. [START_REF] Carlsten | On the instability of bitcoin without the block reward[END_REF],

the authors envision the stability of the system when the block reward reduces to the transaction fees.

Möser & Böhme [START_REF] Möser | Trends, tips, tolls: A longitudinal study of Bitcoin transaction fees[END_REF] analyse the main drivers of the fees and conclude that higher fees lead to faster transaction processing. Easly et al. [START_REF] Easley | From mining to markets: The evolution of bitcoin transaction fees[END_REF] link the proportion of zero-fee transactions to the bitcoin price and memory pool size through a linear model. Tedeschi et al. [START_REF] Tedeschi | On optimizing transaction fees in Bitcoin using AI: Investigation on miners inclusion pattern[END_REF] build a neural net that outputs the probability for a transaction to be included based on the transaction features. Finally, Rossi et al. [START_REF] Rossi | Analysis of the confirmation time in proof-ofwork blockchains[END_REF] consider a queueing model to estimate the confirmation time.

The block reward is expressed in crypto-currency units, but the operational cost in (1) is likely to be expressed in fiat currency. The question of modelling the exchange rate of crypto against fiat currencies naturally arises when studying the profit and losses of blockchain miners via model [START_REF] Albrecher | Blockchain mining in pools: Analyzing the trade-off between profitability and ruin[END_REF].

The evolution price of cryptocurrencies has been extensively studied in the literature. Ciaian et al. [START_REF] Ciaian | The economics of BitCoin price formation[END_REF] study the bitcoin price formation incorporating market information, such as the Dow Jones stock market index or the oil price. Bouoiyour et al. [START_REF] Bouoiyour | What drives Bitcoin price[END_REF] decompose the bitcoin price index using Empirical Mode Decomposition, which is similar to signal-processing techniques, but does not assume periodicity, see e.g. [START_REF] Zeiler | Empirical mode decomposition-an introduction[END_REF]. Many authors have applied neural network techniques to fit and predict bitcoin prices. For instance, Almeida et al. [START_REF] Almeida | Bitcoin prediciton using ANN[END_REF] use Artificial Neural Networks (ANN) and find that trading volumes are irrelevant. McNally et al. [START_REF] Mcnally | Predicting the price of bitcoin using machine learning[END_REF] use Recurrent Neural Networks (RNN) of Long Short Term Memory type to accomododate their three-year long dataset. Time series models like GARCH [START_REF] Katsiampa | Volatility estimation for bitcoin: A comparison of GARCH models[END_REF] and ARIMA [START_REF] Azari | Bitcoin price prediction: An ARIMA approach[END_REF][START_REF] Made Wirawan | Short term prediction on bitcoin price using ARIMA method[END_REF] have also been considered.

We will pursue two modelling strategies in this paper. First, we assume that the block rewards are independent and identically distributed (i.i.d.) random variables with a generalized hyperexponential (GH) distribution, studied in Botta et al. [START_REF] Botta | Characterizations of generalized hyperexponential distribution functions[END_REF], also referred to as the combination of exponentials model by Dufresne [START_REF] Dufresne | Fitting combinations of exponentials to probability distributions[END_REF]. The probability density function of a GH distribution is a linear combination of exponentials which does not need to be convex (in contrast to mixture of exponential distributions).

The GH class is a proper subset of matrix-exponential (ME) distributions which are probability distributions with rational Laplace transform, see already Cox [START_REF] Cox | A use of complex probabilities in the theory of stochastic processes[END_REF] and Bladt & Nielsen [START_REF] Friis | Matrix-Exponential Distributions in Applied Probability[END_REF] for a recent overview. The GH class leads to tractable calculations and is itself already dense in the set of probability distributions on the positive half-line. For applications in insurance risk theory, see e.g. Lin & Willmot [START_REF] Lin | Analysis of a defective renewal equation arising in ruin theory[END_REF][START_REF] Lin | The moments of the time of ruin, the surplus before ruin, and the deficit at ruin[END_REF]. In that framework, ( 1) is called the dual model of the standard insurance risk model, as the wealth process performs upward jumps and decreases linearly in time. In our previous work [START_REF] Albrecher | Blockchain mining in pools: Analyzing the trade-off between profitability and ruin[END_REF], we found closed-form expressions for the ruin probability and the expected profit when the rewards are GH distributed. In this paper we want to go one step further and fit such a distribution to actual bitcoin data.

Since the construction is not probabilistic, a priori one can not guarantee that a particular fitted set of parameters for a GH (and more generally ME) distribution represents a proper probability distribution. Conditions based on the roots and poles of the Laplace transform have been given in the works of Bean et al. [START_REF] Botta | Characterizations of generalized hyperexponential distribution functions[END_REF] and Fackrell [START_REF] Fackrell | An alternative characterization for matrix exponential distributions[END_REF]. These conditions apply to ME distributions and were integrated in Frackrell [START_REF] Fackrell | Fitting with matrix-exponential distributions[END_REF] within a maximum likelihood estimation procedure. Dufresne [START_REF] Dufresne | Fitting combinations of exponentials to probability distributions[END_REF] used GH distributions to approximate any distribution on the positive half-line via an expansion in terms of Jacobi polynomials. The non-negativity problem is addressed then by expanding the square root of the probability density. In this work, we explore the polynomial approach to fit data to a GH distribution.

The non-negativity of the resulting probability density function is checked via a bisection procedure suggested in the work of Hanzon & Holland [START_REF] Hanzon | Non-negativity analysis for exponential-polynomialtrigonometric functions on [0,∞)[END_REF]. To the best of our knowledge, this paper is the first to fit combinations of exponentials to actual data in a non-parametric way instead of approximating a predefined distribution. The results may therefore also be applicable in other modelling contexts where such a distributional assumption is assumed, see e.g. [START_REF] Albrecher | A direct approach to the discounted penalty function[END_REF] for an example in ruin theory.

The assumption of stationarity, a key premise in our first approach, might be considered too restrictive, as block rewards are influenced by transaction fees and fluctuations in cryptocurrency prices.

Consequently, we adopt a second approach that views block reward data as an ARIMA time series.

After fitting it to the data, we generate scenarios that enable us to estimate risk and performance indicators through Monte Carlo simulations. We then conduct a numerical comparison between the results obtained under this time series framework and the ones based on the i.i.d. assumption in our first approach, together with a series of further sensitivity results on various assumptions underlying the model for which a tractable formula for the key quantities is available.

The rest of the paper is organized as follows. Section 2 provides a brief description of our data together with reminders about the way that Proof-of-Work blockchains operate. Section 3 describes the combination of exponentials model and our block reward distribution function estimators. The proposed estimation is first back-tested on synthetic data, before it is applied to the actual block reward data. Section 4 presents the result of our time series analysis. Section 5 compares the two modelling approaches, looking at their impact on the profitability and ruin of blockchain miners. It also contains sensitivity tests with respect to the inclusion of transaction fees in the modelling as well as the electricity price. Section 6 concludes.

Transaction fee concepts and descriptive data analysis

In this section, we will analyse empirical data on the transaction fees in the bitcoin cryptocurrency.

Let us first give a short reminder on the definition of transaction fees and their importance in the mining process.

In the bitcoin Proof-of-Work verification algorithm, each miner solves a cryptoproblem in order to validate a block. Whenever a block is validated or mined, the miner receives the corresponding block reward set to 6.25 BTC at the time of writing. In addition, the sum of transaction fees attached to all the transactions included in this block are also given as an additional reward to the miner. As part of the validation system, the miner obtains the authorization from the community to choose which transactions from the memory pool ("waiting line") are entering the newly mined block. The user will typically attach a transaction fee to the required payment. This fee mostly depends on the transaction volume, since a block has a fixed allotted space inside. It also depends on the market congestion and the individual user's decisions [START_REF] Antonopoulos | Mastering Bitcoin: unlocking digital cryptocurrencies[END_REF]. There exist algorithms that help the user choose an appropriate fee. For example on https://privacypros.io/tools/bitcoin-fee-estimator/, one can estimate the expected confirmation time as a function of the chosen fee. In the mining process, the miners have to decide which transactions to incorporate in the block, since it has limited memory space. They will consider the priority of the transaction based on the attached fee size per byte, but recent research

shows that there may be other (selfish) interests in promoting transactions [START_REF] Messias | Selfish & opaque transaction ordering in the Bitcoin blockchain: the case for chain neutrality[END_REF]. It is important to note again that with the scheduled halving of the fixed block rewards every four years, the share of the transactions fees in the total rewards will gain decisive importance over time. Thus, including and modelling this stochastic part will become even more relevant in the future.

For this analysis, we use publicly available data. Scrapping bitcoin-related data is possible by making calls to some APIs. This was done by looping through necessary blocks. For example, to access block number 650000 information, one can follow the address https://chain.api.btc.com/v3/block/ 6500001 . It extracts information in JSON format, which can then be reformatted to our needs. With this method, we gather data for the period spanning from the last halving of the bitcoin reward on May 12, 2020 until September 16, 2021. For this time frame, we collect the following information:

• The transaction fees per block;

• The exchange rate BTC-USD per minute;

• The current difficulty of the cryptoproblem (adjusts bi-weekly);

• The size of the memory pool of the transactions in bytes (daily);

• The number of transactions in the memory pool (daily).

Figure 1 depicts the extracted data and Table 1 contains some statistics of the dataset. One can observe a strong correlation between the transaction fees and the number of transactions in the memory pool, which is confirmed in Table 2. For the analysis in this paper, we consider a smaller sub-sample of the data containing 10'000 data points ranging from February 10, 2021 to April 21, 2021. Also, we merge the series of fees and prices by converting fees in USD, since we aim to model the latter. The covariate "difficulty" is not retained as it has a bi-weekly update and our focus will be on a two weeks horizon. For this paper, we also do not further consider the strongly correlated variables "mempool size" and "mempool count", because fitting a time series with a covariate implies modelling the dynamics of this covariate independently.

-0 5 -1 2 2 0 2 0 -0 7 -0 5 2 0 2 0 -0 8 -2 9 2 0 2 0 -1 0 -2 3 2 0 2 0 -1 2 -1 7 2 0 2 1 -0 2 -0 9 2 0 2 1 -0 4 -0 5 2 0 2 1 -0 5 -3 0 2 0 2 1 -0 7 -2 4 2 0 2 1 -0 9 -1 6 Date (e) BTC/USD price 0 50000 100000 150000 Fees in USD 2 0 2 0 -0 5 -1 2 2 0 2 0 -0 7 -0 5 2 0 2 0 -0 8 -2 9 2 0 2 0 -1 0 -2 3 2 0 2 0 -1 2 -1 7 2 0 2 1 -0 2 -0 9 2 0 2 1 -0 4 -0 5 2 0 2 1 -0 5 -3 0 -0 7 -2 4 2 0 2 1 -0 9 -1 6 Date (f) Fees in USD
For the purpose of this paper, we opt for modelling only one time series only, the dynamics of which then aggregates all the dynamics playing a role for the output variable. As illustrated in Figure 2, the time series of fees exhibits strong autocorrelation, so that a stationarity assumption would not be appropriate. However, by taking the first difference, one can find back stationarity to a large extent, see the right-hand side of Figure 2; only one lag is still significant for the fees. We will therefore consider only one lag when fitting a time series model in Section 4.

Block reward as a combination of exponentials

A random variable U has a generalized hyperexponential distribution if its cumulative distribution function (CDF) is given by

F U (x) = 1 - d i=1 a i e -λ i x , for x ≥ 0, (2)
where λ 1 , . . . , λ d > 0 and a 1 , . . . , a d ∈ R with d i=1 a i = 1. In the sequel, we assume that λ 1 < . . . < λ d . Define the vectors

λ = (λ 1 , . . . , λ d ) , a = (a 1 , . . . , a d ) ,
and the diagonal matrix Λ = diag(λ), so that

F U (x) = 1 -a • e -Λx • 1 d , for x ≥ 0, (3)
where

1 d = (1, . . . , 1)
. The probability density function (PDF) is given by

f U (x) = b • e -Λx • 1 d , for x ≥ 0, (4)
with b = a • Λ.

The Laplace transform of U

E(e -θU ) = b • (Λ + θI d ) • 1 d , θ ≥ 0, (5)
where I d is the identity matrix, is rational which implies that combinations of exponential distributions are instances of matrix-exponential distributions.

Non-negativity of GH probability distribution functions

Given a set of parameters a and λ, there is no straightforward way to ensure that ( 3) is a proper CDF or that ( 5) is a proper PDF. Some characterization, based on the Laplace transform, have been provided by Bean et al. [START_REF] Bean | Characterization of matrix-exponential distributions[END_REF] and Fackrell [START_REF] Fackrell | An alternative characterization for matrix exponential distributions[END_REF] for ME distributions. We take a different road here, following up on the work of Hanzon and Holland [START_REF] Hanzon | Non-negativity analysis for exponential-polynomialtrigonometric functions on [0,∞)[END_REF]. Consider the function

f (x) = b • e -Λx • 1 d , x ≥ 0. (6)
In order for f to be a proper PDF, we need to ensure f (x) > 0, ∀x > 0, and

∞ 0 f (x)dx = 1. (7)
Necessary conditions for [START_REF] Bean | Characterization of matrix-exponential distributions[END_REF] to hold include b 1 > 0 and b • Λ -1 = 1 d , but the latter are not sufficient.

We therefore use a verification method via bisection, suggested in Hanzon and Holland [START_REF] Hanzon | Non-negativity analysis for exponential-polynomialtrigonometric functions on [0,∞)[END_REF]. We define the sequence

f 0 (x) = b • e -Λx • 1 d = f U (x) and f k (x) = b • k i=1 (λ j • I d -Λ) • e -Λx • 1 d , for k = 1, . . . , d.
It is characterized by the following property: for x ∈ [0, M ], the function f k has at most one signchanging zero between two sign-changing zeros or boundary points of f k+1 for k = 0, 1, . . . , d -1. As

f d (x) = 0, one can recursively, starting from f d (x)
, check the presence of sign-changing points on a closed interval through a bisection procedure.

Fitting GH distributions to data via polynomial expansions

Dufresne [START_REF] Dufresne | Fitting combinations of exponentials to probability distributions[END_REF] presents a method to approximate any PDF of a distribution on the positive half-line via a combination of exponentials. The approximation formula takes the form of an expansion in terms of the shifted Jacobi polynomials defined as

R (α,β) k (x) = k j=0 ρ k,j x j , ( 8
)
where

ρ k,j = (-1) k (β + 1) k (-k) j (k + α + β + 1) j (β + 1) j k!j! , ( 9
)
where (z

) k = z • (z + 1) • (z + 2) • • • • • (z + k -1)
(x) = (1 -x) α x β .
Any function g : (0, 1) → R, square integrable w.r.t. ϕ(x) can be expanded as a shifted Jacobi polynomial expansion with

g(x) = ∞ k=0 c k R (α,β) k (x), (10)
where

c k = 1 h k 1 0 g(x)(1 -x) α x β R (α,β) k (x)dx, (11)
and

h k = 1 0 (1 -x) α x β R (α,β) n (x) 2 dx = Γ(n + α + 1)Γ(n + β + 1) (2n + α + β + 1)n!Γ(n + α + β + 1) . ( 12
)
The convergence in [START_REF] Carlsten | On the instability of bitcoin without the block reward[END_REF] takes place in the L 2 sense, see for instance the book of Nagy [START_REF] Szeokefalvi-Nagy | Introduction to real functions and orthogonal expansions[END_REF]Ch.7].

Our target is a PDF f on the positive half-line. Following Dufresne [START_REF] Dufresne | Fitting combinations of exponentials to probability distributions[END_REF], we expand the function f * (x) = e prt f (x) for some p ∈ R and r > 0 and use the change of variable

g(x) = f * - 1 r log(x) ,
that maps the interval (0, 1) onto (0, ∞). The expansion of f is then

f (x) = e -rpt ∞ k=0 c k R (α,β) k (e -rpt ), (13)
and the square integrability condition on g translates directly to

∞ 0 e -(β+1-p)rt (1 -e -rt ) α f 2 (t)dt < ∞.
The coefficients of the polynomial expansion can also be expressed as an integral in terms of f as

c k = r h k ∞ 0 e -(β-p+1)rt (1 -e -rt )R (α,β) n (e -rt )f (t)dt. (14)
A simple truncation of the infinite series in (13), followed by a normalization so that it integrates to 1, yields an aproximation

f (t) ≈ e -rpt d-1 k=0 c k R (α,β) k (e -rt ), (15)
which is consistent with what is referred to as Method A in Dufresne's work [START_REF] Dufresne | Fitting combinations of exponentials to probability distributions[END_REF]. Replacing the polynomials R

(α,β) k (x) by k j=0 ρ k,j x j in (17) yields f (t) ≈ d j=1 d-1 k=j-1 c k ρ k,j-1 e -(j-1+p)rt ,
which is a combination of exponentials as in [START_REF] Antonopoulos | Mastering Bitcoin: unlocking digital cryptocurrencies[END_REF] where

b := d-1 k=0 c k ρ k,0 . . . c d-1 ρ d-1,d-1
and λ := pr (1 + p)r . . . (n + p)r .

Remark 3.1. The approximation method involves selecting parameters α, β, p, r, and determining the truncation order d. In the absence of established selection guidelines, we draw upon the parameter values utilized in Dufresne's work [START_REF] Dufresne | Fitting combinations of exponentials to probability distributions[END_REF]. The choice of the truncation order should strike a balance between accuracy, computational efficiency, and numerical stability, favoring larger values wherever possible. When considering f * instead of f , it ensures that f (t) approaches zero as t tends towards infinity for any truncation order, provided that 0 < p < (β + 1)/2.

For our application, we do not have a known distribution function to approximate, but a dataset to fit.

Assume that {x 1 , . . . , x M } form an i.i.d. sample of size M . We can replace the expansion coefficients defined in [START_REF] Easley | From mining to markets: The evolution of bitcoin transaction fees[END_REF] by their empirical counterpart with

c k = r h k M M m=1 e -(β-p+1)rxm (1 -e -rxm )R (α,β) k (e -rxm ), for k = 0, . . . , d -1. ( 16
)
An a posteriori control with the help of the bisection method from Section 3.1 can ensure the nonnegativity of the estimated PDF

f M (t) = e -rpt d-1 k=0 c k R (α,β) k (e -rt ). ( 17
)
The estimated PDF is a nonparametric density estimator relying on orthogonal functions, a method detailed in [START_REF] Wasserman | All of nonparametric statistics[END_REF]Ch.8]. A recognized limitation of this approach is its susceptibility to occasional negative values stemming from sampling errors. In instances where our estimates exhibit negativity,

we can employ what Dufresne [START_REF] Dufresne | Fitting combinations of exponentials to probability distributions[END_REF] terms as 'Method B' as a corrective measure. Instead of expanding e prt f (t), consider expanding f (t) = e prt f (t).

We get an approximation formula of the form

f (t) ≈ e -prt d-1 k=0 c k R (α,β) k (e -rx ) = d j=1 b j e -λ j t , ( 18
)
and finally squaring it yields an approximation of f which is a proper PDF after normalization:

f (t) ≈ d j=1 d k=1 b j b k e -(λ j +λ k )t = 2d-
b k = r h k ∞ 0 e -(β-p+1)rt (1 -e -rt )R (α,β) k (e -rt ) f (t)dt. ( 19
)
To get a statistical estimation of the coefficients, we replace f in ( 19) by a kernel density estimator

f h (x) = 1 M M m=1 K h (x -x m ) = 1 M h M m=1 K x -x m h , ( 20
)
where K(x) is the Gaussian kernel.

Remark 3.2. In contrast to Method A, Method B guarantees a valid PDF. However, this advantage comes at the expense of significantly increased computational complexity due to doubling the number of terms (with a numerical integration of the kernel density estimator for each).

Simulation study

We illustrate our fitting procedure for a combination of exponentials through a brief simulation study.

Draw m samples (x 1 , . . . , x m ) from a right-shifted gamma random variable X = γ + Y , where γ > 0 and Y has PDF

f (x) = δ r x r-1 e -δx Γ(r) , x > 0, r, δ > 0. ( 21
)
The choice of this shift is motivated by the observed shape of the empirical distribution in our collected block reward data later. We set the parameters to r = 3, δ = 0.5 and γ = 5. Figure 3 shows the fit (in red) of the combination of exponentials distribution to the data when using Method A with parameters α = 0, β = 0, r = 0.01, p = 0.9, d = 20 for samples of sizes n ∈ {100, 1000, 10000, 100000}.

The quality of the fit enhances as the sample size increases. However, note that also, for larger sample N = 20, α = 0, β = 0, r = 0.01, p = 0.9 results in a valid probability distribution. Due to this essential property, we have chosen to exclusively employ Method B in our application to the block reward data in the sequel.

Real data application

The block reward comprises two components: the reward for discovering a new block, which currently stands at BTC6.25 at the time of writing, and the transaction fees detailed in Section 2. Given that miners typically operate within a fiat currency framework, such as USD, we apply the exchange rate applicable at the moment of block discovery. polynomial expansion, provides an advantage by capturing variations in the central mass, even in the absence of data points in the lower tail.

Figure 6 shows the fit of the combination of exponentials using Method B with two parameterizations.

Block rewards as time series

Since we also want to test the model for its sensitivity to non-stationarities, we fit the block reward data to an ARIMA model calibrated using the Box and Jenkins optimization method. Recall that a time series X t is ARIMA(p, d, q) if ∇ d X t is an ARMA(p, q) process, where ∇ d is the d th difference

operator. An ARMA(p, q) time series is a stationary process defined as

X t = ϕ 1 X t-1 + • • • + ϕ p X t-p + z t + θ 1 z t-1 + • • • + θ q z t-q , ( 22
)
with ϕ p ̸ = 0, θ q ̸ = 0 and z t is white noise with mean 0 and variance σ 2 z , see e.g. [START_REF] Robert H Shumway | Time series analysis and its applications[END_REF]. Note that time series modeling is not a focus of this paper, we refer to other papers for this purpose, see e.g. [START_REF] Azari | Bitcoin price prediction: An ARIMA approach[END_REF][START_REF] Made Wirawan | Short term prediction on bitcoin price using ARIMA method[END_REF].

Our choice of a simple ARIMA model is motivated by its suitability for generating plausible scenarios over a short time horizon while departing from the i.i.d. assumption of Section 3. Also, we opted against more complex GARCH models due to their tendency to overfit the data and their requirement for an extensive number of lags to ensure reliability. Hence, we deliberately select a simple ARIMA models as a pragmatic and reliable choice to conduct our risk analysis over a two weeks time horizon.

For fitting our model, we consider a data set from a time frame between February 10, 2021 and April 6, 2021 that we split into a training set and a consecutive test set for checking the data prediction.

In Table 3 we summarize the obtained results. The fitted model suggests indeed a once differentiated series. In Figure 7 we show an illustration of the fitted model. On the left-hand side, one can see the In addition to the statistical fit (cf. Figure 8 for a normal Q-Q plot for the remaining residuals), the fit also seems quite satisfactory visually, which is remarkable for the case of only a few free parameters.

Comparison of the two modelling approaches in terms of risk

In [START_REF] Albrecher | Blockchain mining in pools: Analyzing the trade-off between profitability and ruin[END_REF][START_REF] Albrecher | On the profitability of selfish blockchain mining under consideration of ruin[END_REF], explicit formulas were derived for the ruin probability and the expected surplus of a miner, taking into account the option of participating in a mining pool. Assume the miner's surplus has the form

R i t = u -c i • t + Nt i=1 U i , t ≥ 0,
where N t ∼ P oisson(p i µt) and the U i 's are i.i.d. with PDF f U (x) = d j=1 a j λ j e -λ j x , x > 0. In [START_REF] Albrecher | Blockchain mining in pools: Analyzing the trade-off between profitability and ruin[END_REF], it was shown that under the assumption of i.i.d. rewards of GH type, the ruin probability up to an exponential time horizon (with mean t) can be expressed as ψ(u, t) = e -R•u , [START_REF] Messias | Selfish & opaque transaction ordering in the Bitcoin blockchain: the case for chain neutrality[END_REF] and the miner's expected value of the surplus at that exponential time horizon, given that it did not go negative until then, is given by

V (u, t) = t   c i -p i µ d j=1 a j λ j   e -Ru + u + t   p i µ d j=1 a j λ j -c i   , u > 0, ( 24
)
where R is the unique solution with positive real part of the equation

c i R + p i µ d j=1 a j λ j R + λ j - 1 t + p i µ = 0.
These metrics are of particular significance, given that the process of mining incurs considerable energy costs, and real-world miners may face the risk of financial ruin, which in turn affects their expected earnings.

Our goal in this section is to assess the sensitivity of the risk measures for which we have formulas ( 23) and ( 24) w.r.t. some of the model assumptions. First, in Section 5.1 we compare the formulas (using a GH fit to block rewards data from the time period February 10, 2021 to April 21, 2021) to the actual historical realization of the occurrence of blocks and the sizes of the rewards in the period from February 10, 2021 to a random time horizon in the future with a mean of 2 weeks. Note that there are still three random elements to be implemented in the historical path: the actual length of the random time horizon and the probability that a found block was found by a particular miner. Thirdly, we consider a 'Full Pay-per-Share' reward system (FPPS) for the pool, in which pool managers instantly reward miners for each share submitted, with the payout determined by the block reward and the entire estimated transaction fees associated with the block. That is, we have to simulate the arrival of shares in addition to the arrival of blocks. As the historical sample path both contains potential time-dependence of transaction fees and the actual block reward values (rather than the GH fit), but only one realization of the two latter effects, we subsequently compare in Section 5.2 the results of ( 23) and ( 24) with the simulated counterparts under an i.i.d. assumption, but bootstrapping block reward sizes from the empirical distribution function over the period February 10, 2021 to April 21, 2021. Subsequently, in Section 5.3 we provide a comparison of the formulas ( 23) and ( 24) with the simulated counterparts under the ARIMA assumption calibrated in Section 4. Finally, Sections 5.4 and 5.5 consider the sensitivity of the model output concerning the inclusion of transaction fees at all, as well as concerning the price of electricity.

Comparison with the historical path

With the collected data on the fees and prices, we can reconstitute the real surplus path of miners or pools in any specific past time period. Indeed, if we position ourselves at some starting date, we can replicate the outflows of mining costs and the inflows of block rewards for the individual miner as well as for the pool. For the following example, we select a two weeks time frame. We select the documented block arrival in the entire system to the particular pool by simulating a Bernoulli random variable with probability equal to the pool's proportion of computational power in the global mining network. For the individual miner, we also simulate the more frequent share payouts by assuming

Poisson distributed arrivals of their rewards. In Figures 9 and10, one can see an illustration of the surplus path for the pool and the miner, respectively. Figures 9a and9b are almost indistinguishable to the naked eye, but they are not identical. Indeed, for that short particular time horizon, the exchange rate was not very volatile. Here we choose the same parameters as in [START_REF] Albrecher | Blockchain mining in pools: Analyzing the trade-off between profitability and ruin[END_REF]: t = 336h = 2 weeks, p i = 0.001, q = 0.1, f = 0.02, µ = 6p i = 0.006, u = $1M , the cost of electricity c i is given by c = p i × e W × π W , where e W is the electricity consumption of the network expressed in kWh, and π W = 0.04 is the price of electricity per kWh. For e W , we choose 115.541×10 9 365.25×24 . Our analysis distinguishes between scenarios where the miner is engaging in solo mining or participating in a pool. More specifically, we implement the following approach:

1. We choose February 10, 2021 as the starting date.

2. We randomize the time horizon by simulating nsim durations following an exponential distribution with mean equal to 2 weeks.

3. For each simulation run, we simulate nsim share reward payment times.

4. For each block found by an individual miner, mining alone, he will receive the block reward and the transaction fees attached to this block. For a miner in a pool, the miner receives only a fraction of the block reward and of the transaction fees, assuming a FFPS pooling scheme.

In practice, whereas the bitcoin block reward is known and fixed (at 6.25 BTC at the time of writing), the transaction fees included in a block can only be discovered after the block is appended to the blockchain. It means that the pool has to predict the future fee in order to pay the miners before the block appears. What often happens in practice, is that the fee is computed as the average fee over some short time frame, e.g. the last 24 hours [START_REF] Tech | Different Bitcoin mining pool payment methods PPS vs FPPS vs PPLNS vs PPS+)[END_REF]. This is the retained approach in our framework.

5. Combine all data to produce sample paths of the miner's surplus for different values of their initial capital u. Averaging over all iterations yields simulated values for V (u, t) and ψ(u, t).

On the other hand, we compute ( 23) and ( 24) under the i.i.d. assumption with a GH model for the block rewards as calibrated in Section 3, using the moderate truncation parameter d = 10 (a higher number of terms in the combinations of exponentials would significantly complicate the numerical evaluation).

Figures 11 and12 display the ruin probability and expected surplus based on the theoretical formulas and the historical path method, for both a stand-alone miner and one joining a pool. One can see that the results are rather close for the two cases, and the shape as a function of initial capital u is identified rather well. This suggests that the theoretical formulas are quite useful for practical use.

Sensitivity of the GH fit under the i.i.d. assumption

Under the assumption of i.i.d. block rewards, it is interesting to see how sensitive the results are w.r.t.

to the fitted GH model. For that purpose, we compare the formulas [START_REF] Messias | Selfish & opaque transaction ordering in the Bitcoin blockchain: the case for chain neutrality[END_REF] and [START_REF] Friis | Matrix-Exponential Distributions in Applied Probability[END_REF] to a situation, where we resample block reward values from the empirical distribution function of rewards from the period February 10, 2021 to April 21, 2021. In a sense, this may also be considered a fairer comparison than the one in Section 5.1, since now the variability of block rewards is similar. Figures 13 and14 depict the results, where the black lines represent the formulas using the GH fit, and the red lines are simulated values under the resampling together with a 95% confidence interval (also in all the remaining plots of this section, whenever we plot simulated values we do so together with its 95% confidence interval). Since this data sample served as the basis for the GH fit calibration, one expects a close similarity, which is indeed the case. Consequently, the GH approximation seems sufficient for purposes of drawing conclusions for our quantities of interest.

Sensitivity w.r.t. time dependence of rewards

We use the ARIMA model calibrated in Section 4 to simulate trajectories for the transaction fees, as well as another ARIMA model for the bitcoin price in USD relevant for the fixed 6.25 BTC contributions (as was suggested by Azari [START_REF] Azari | Bitcoin price prediction: An ARIMA approach[END_REF]). For our time window, a calibration of the latter ARIMA model suggests an ARIMA(5,1,1) model. The sum of the two form our block reward process, which feeds a Monte Carlo estimator of the ruin probability and expected surplus under that assumption.

As previously, we fit the block rewards expressed in USD. For the miner, the price conversion seems interesting, as rewards enter less often, therefore the timing of the conversion is important and taken in account. From the pool perspective, it virtually operates in BTC for both inflows and outflows and therefore does not suffer conversion risk. Figure 15 and 16 compare the results (in red) with the formulas ( 23) and ( 24) (in black), which were derived under an i.i.d. assumption. One observes that for the ruin probability, the deviation from the i.i.d. assumption observed in the data is not relevant for the 2 weeks timeframe, and also for the expected value of the future surplus, the deviations are minor. The difference in the expected profit for large initial capital (where ruin becomes very unlikely) can be explained by the fact that for the calibration of the ARIMA model the time period is shorter (by 20%) compared to the one of the i.i.d. GH fit due to out-of-sample validation, cf. Section 4.

Sensitivity w.r.t. transaction fees

Transaction fees represent an additional element of randomness in the block rewards, adding to the inherent volatility of the bitcoin (fiat) price. Our objective is to illustrate the effects of including or excluding transaction fees in the reward modelling process. Figures 17 and18 depict the changes in the ruin probability and expected surplus for miners, both solo and within a pool, when transaction fees are inlcuded or left out (that is, only the ARIMA model for the bitcoin price is simulated in the latter case). One observes that the expected surplus converges to different limits as the initial capital u grows large. In that case, the impact of ruin on the miner's surplus diminishes, and only the positive effect of higher block rewards materializes (statistically, for the period May 12, 2020 to September 16, 2021, the transaction fees mounted to 8.6% of the fixed part of the block rewards 6.25 BTC per bounty). In terms of ruin probabilities, while the model including fees prevails over the one without fees, the difference is in fact very small. However, it is important to consider that the fixed block reward undergoes a scheduled halving approximately every four years. According to the current countdown3 , the next halving is expected in April 2024, reducing the fixed reward to 3.125 BTC. In Figures 19 and20, we therefore conduct the same analysis with 3.125 BTC for the fixed part, all other factors held constant (and assuming that the fee dynamics remain unchanged after that halving). In the latter case, the differences become much more pronounced.

Sensitivity w.r.t. electricity costs

Finally, let us consider the sensitivity of the results to electricity costs under the ARIMA model.

Figures 21 and22 show the ruin probability and expected surplus as a function of the price of electricity for a solo miner and a mining pool participant for a fixed initial capital of u = 100 ′ 000. Such a graph can help to identify upper bounds for affordable electricity prices needed to ensure specific target levels of ruin probability or expected surplus for a given level of u.

Notably, the results reveal a substantial difference in the ruin probability between individual miners and those participating in a pool, emphasizing the risk mitigation benefits of pooling (already observed in [START_REF] Albrecher | Blockchain mining in pools: Analyzing the trade-off between profitability and ruin[END_REF]). Additionally, this quantifies the influence of increased electricity costs, underlining the exposure and vulnerability of 'Proof-of-Work' cryptocurrencies to energy-related crises. Such considerations may (even economically) motivate the transition to alternative, less energy-intensive consensus protocols.

For instance, Ethereum, the second-largest cryptocurrency, shifted from 'Proof-of-Work' to 'Proof-of-Stake' on September 15, 2022, reducing its energy consumption by a remarkable 99.95%.

Conclusion

In this paper, we have delved deeper into the framework for analyzing bitcoin mining from the perspective of risk and profitability. Previously, quantifying the choices available to miners, such as entering a mining pool and selecting the most suitable one, relied on formulas based on assumptions involving combinations of exponential distributions. In this work, we introduce a straightforward and efficient approach for fitting real-world data to these distributions, allowing us to apply theoretical results in practical mining scenarios. The remarkable flexibility of the resulting method enables us to explore shapes that are typically challenging to achieve using other approaches. The code for this work is available upon request.

Furthermore, we have explored the stochastic nature of block rewards, breaking down their variability into two key components: price volatility and the inclusion of transaction fees. Our findings highlight the growing importance of incorporating fees in modeling, especially with the scheduled halving of the fixed block reward.

By expressing transaction fees as a time series, we can simulate our key metrics of interest-namely, the expected surplus and ruin probability-and compare them to theoretical results. This analysis confirms that the formulas derived in [START_REF] Albrecher | Blockchain mining in pools: Analyzing the trade-off between profitability and ruin[END_REF][START_REF] Albrecher | On the profitability of selfish blockchain mining under consideration of ruin[END_REF], originally under an i.i.d. assumption, stand up well when compared to the outcomes derived from modeling strategies that incorporate time dependency or use empirical data more generally. Finally, we emphasize the sensitivity of our key metrics to the inclusion of transaction fees. While the differences in magnitude are currently still small, we anticipate significant shifts in results after the next halving, making fees a more substantial factor in modeling. Our analysis also underlines the substantial impact of electricity costs on a miner's profitability and ruin probability. These insights contribute to the ongoing debate surrounding the viability of 'Proof-of-Work' consensus protocols in an increasingly energy-constrained world.

Figure 1 :

 1 Figure 1: Illustration of the data

Figure 2 :

 2 Figure 2: Autocorrelation function of fees in USD (left) and its first differences (right)

1 m=1

 1 bm e -(m-1+2p)rt , with bm = m j=1 b j b m+1-j and b j>d := 0, since λ j = (j -1 + p)r, j = 1, 2, . . . , d. The coefficients of the polynomial expansion of √ f are given by

14 ( 14 (

 1414 Sample size: 100. N = 20, α = 0, β = 0, r = 0.01, p = 0Sample size: 1000. N = 20, α = 0, β = 0, r = 0.01, p = 0c) Sample size: 10000. N = 20, α = 0, β = 0, r = 0.01, p = 0d) Sample size: 100000.

Figure 3 :

 3 Figure 3: Fitting of the shifted Gamma random sample by modified Method A.

Figure 4

 4 Figure4shows the fit (in red) of the combination of exponentials distribution to the data when using Method B with parameters α = 0, β = 0, r = 0.01, p = 0.9, d = 20, for samples of sizes n ∈ {100, 1000, 10000, 100000}. The fit is less good when using Method B; however, it consistently

Figure 4 :

 4 Figure 4: Fitting of the shifted Gamma random sample by modified Method B.

Figure 5

 5 depicts the histogram of total rewards received by miners, presented both in BTC and USD. The data is indeed shifted away from zero due to the fixed block reward addition, which poses a challenge for parametric fitting methods, as the support of exponential random variables is the entire positive halfline. The non-parametric approach, utilizing a Rewards expressed in USD.

Figure 5 :

 5 Figure 5: Histogram of total rewards, period from February 10, 2021 to April 21, 2021.

Figure 6 :

 6 Figure 6: Approximation of the bitcoin rewards sample (Method B, α = 0, β = 0, r = 0.0000008)

  Simulated rewards per block in USD.

Figure 7 :

 7 Figure 7: ARIMA(1,1,7) fit to block reward data

Figure 8 :

 8 Figure 8: Normal Q-Q plot of ARIMA(1,1,7) residuals.

Figure 10 :

 10 Figure 10: Miner surplus path USD

  Expected value of the surplus.

Figure 11 :

 11 Figure 11: Simulation of a historical path vs i.i.d. GH fit, miner in a pool (starting date February 10, 2021).

  Expected value of the surplus.

Figure 12 :

 12 Figure 12: Simulation of a historical path vs i.i.d. GH fit, miner alone (starting date February 10, 2021)

  Expected value of the surplus.

Figure 13 :

 13 Figure 13: Empirical reward distribution vs i.i.d. GH fit, miner in a pool (starting date February 10, 2021).

  Expected value of the surplus.

Figure 14 :

 14 Figure 14: Empirical reward distribution vs i.i.d. GH fit, miner alone (starting date February 10, 2021).

  Expected value of the surplus.

Figure 15 :

 15 Figure 15: ARIMA simulation vs i.i.d. GH fit, miner in a pool (starting date February 10, 2021)

Figure 16 :

 16 Figure 16: ARIMA simulation vs i.i.d. GH fit, miner alone (starting date February 10, 2021).

4

 4

  Expected value of the surplus.

Figure 17 :

 17 Figure 17: Sensitivity to presence/absence of fees, miner in a pool.

  Expected value of the surplus.

Figure 18 :

 18 Figure 18: Sensitivity to presence/absence of fees, miner alone.

  Expected value of the surplus.

Figure 19 :

 19 Figure 19: Sensitivity to presence/absence of fees, miner in a pool, block reward 3.125 BTC.

Figure 20 :

 20 Figure 20: Sensitivity to presence/absence of fees, miner alone, block reward 3.125 BTC.

Figure 21 :

 21 Figure 21: Sensitivity to change of electricity price π W , miner in a pool (starting date February 10, 2021).

Figure 22 :

 22 Figure 22: Sensitivity to change of electricity price π W , miner alone (starting date February 10, 2021).

Table 1 :

 1 Main statistics of the dataset.

		Block	Date	Fees in BTC Price in USD Mempool size Mempool count Difficulty
	Mean			0.594	29'535	36'090'588	25'610	1.86E+13
	Min	630'014 12.05.2020 0	8'584	120'928	290	1.37E+13
	25 th pct.			0.1731	11'324	3'889'528	4'757	1.68E+13
	Median			0.4652	31'700	27'446'996	18'670	1.86E+13
	75 th pct.			0.8872	46'331	70'698'937	38'942	2.08E+13
	Max	700'851 16.09.2021 4.074	64'617	110'094'398	138'640	2.51E+13

Table 2 :

 2 Pearson correlation matrix.

		Fees in BTC Mempool size Mempool count Difficulty Price
	Fees in BTC	1	0.4885764	0.6361694	0.3313047 -0.01781731
	Mempool size	0.48857638	1	0.7462129	0.6715088 0.43744576
	Mempool count 0.63616939	0.7462129	1	0.5342133 0.25523618
	Difficulty	0.33130472	0.6715088	0.5342133	1	0.54365471
	Price	-0.01781731 0.4374458	0.2552362	0.5436547 1

  denotes the Pochhammer symbol. These polynomials are orthogonal on [0, 1] w.r.t. the weight function ϕ

Table 3 :

 3 Summary of ARIMA model.

		ar1	ma1	ma2	ma3	ma4	ma5	ma6	ma7
		-0.7799	0.0989 -0.6543 -0.1146 -0.0337 -0.0330 -0.0075 -0.0267
	s.e.	0.1122	0.1125 0.0771 0.0194 0.0136 0.0138 0.0123 0.0122
	σ 2	219226983
	Log.Lik. -88159.57
	AIC	176337.1
	BIC	176400

fitted data points in comparison to the historical values of the training sample. On the right-hand side, we show simulated paths in different colors and the true historical values from our test sample in black.

last accessed on 12/09/2023

https://cbeci.org/

See, for example, https://www.nicehash.com/countdown/btc-halving-2024-05-10-12-00.

Source: https://ethereum.org/en/upgrades/merge/ (last accessed on 13/09/2023).