Frederik Bopp

Jonathan Rojas

Natalia Revenga

Hubert Riedl

Friedrich Sbresny

Katarina Boos

Tobias Simmet

Arash Ahmadi

David Gershoni

Jacek Kasprzak

Arne Ludwig

Stephan Reitzenstein

Andreas Wieck

Dirk Reuter

Kai Müller

Jonathan J Finley

Selective optical charge-state preparation of a quantum dot molecule for independent control of orbital coupling

Tunnel-coupled pairs of optically active quantum dots -quantum dot molecules (QDMs) -offer the possibility to combine excellent optical properties such as strong light-matter coupling with twospin singlet-triplet (S -T0) qubits having extended coherence times. The S -T0 basis formed using two spins is inherently protected against electric and magnetic field noise. However, since a single gate voltage is typically used to stabilize the charge occupancy of the dots and control the inter-dot orbital couplings, operation of the S -T0 qubits under optimal conditions remains challenging. Here, we present an electric field tunable QDM that can be optically charged with one (1h) or two holes (2h) on demand. We perform a four-phase optical and electric field control sequence that facilitates the sequential preparation of the 2h charge state and subsequently allows flexible control of the inter-dot coupling. Charges are loaded via optical pumping and electron tunnel ionization. We achieve one-and two-hole charging efficiencies of 93.5±0.8 % and 80.5±1.3 %, respectively. Combining efficient charge state preparation and precise setting of inter-dot coupling allows control of few-spin qubits, as would be required for on-demand generation of two-dimensional photonic cluster states or quantum transduction between microwaves and photons.

I. INTRODUCTION

Long coherence times, strong light-matter coupling and tunability lie at the heart of spin-photon interfaces required for distributed quantum technologies [1]. Semiconductor quantum dots (QDs) provide these characteristics due to their robust polarization selection rules that allow mapping between spin and optical polarization [2][3][4], dominant emission into the zero-phonon line at low temperatures [5], nearly Fourier-limited optical linewidths [6] and integratability into devices to facilitate tunability and enhance single spin-photon coupling efficiencies [7]. Together, these properties make QDs promising as spin-photon interfaces [8] for the on-demand generation of 1D photonic cluster states [9,10] or quantum transduction between microwave and infrared photons [11].

The growth of vertically stacked pairs of tunnelcoupled dots -quantum dot molecules (QDMs) -opens the way to form multi-spin qubits that are less susceptible to decoherence [12]. In particular, the singlet-triplet (S -T 0 ) logical qubit formed by two spins [13][14][15] occupying the hybridized orbitals of tunnel-coupled dots is expected to provide more than an order of magnitude * finley@wsi.tum.de longer coherence times (T ( * )

2 ) due to the existence of a sweet spot at which the S -T 0 qubit energy is insensitive to magnetic and electrical noise. This expectation has been confirmed for both optically active [16] and electrostatically defined QDs [17]. QDMs have also been theoretically suggested to facilitate the deterministic (ondemand) generation of two-dimensional photonic cluster states [18], a key resource needed for measurementbased quantum computation [19] and memory-free quantum communication [20,21]. To facilitate these applications, QDMs must be operated in a regime where they are stably occupied by two spins, while the inter-dot tunnel coupling of s-orbital states can be freely tuned, e.g. using the voltage applied to a gate electrode (V G ). Thus, the charge occupancy of bottom (S B ) and top (S T ) dots in the molecule should remain in the (S B , S T ) ∈ {(2, 0), (1, 1), (0, 2)} subspace and the tunnel coupling of the orbital states should be freely tunable without changing n = S B + S T = 2. In previous experiments, all requirements have been demonstrated individually [22][23][24][25][26]. However, independent control of n and tunnelling induced hybridization of orbital states has been difficult to achieve until now since n is typically regulated via Coulomb blockade of carrier tunneling from a proximal doped contact [16,27] using a single gate electrode. The same gate electrode is also used to tune the orbital states into resonance, the proximity of the QDM to the doped arXiv:2205.10001v1 [cond-mat.mes-hall] 20 May 2022 contact, dot-dot spacing (s) and height (h) of the two dots forming the molecule must be precisely controlled during growth to allow for tunnel coupling between the s-orbital states in the QDM while remaining in the n=2 charge stability region.

Here, we demonstrate all-optical, sequential and independent preparation of the n = 1 and n = 2 charge states of the QDM in a device geometry that leaves the gate potential free to control the orbital tunnel coupling between the two dots. Our device geometry is an n-i-Schottky diode with the QDMs embedded at the midpoint of the i-region (see Experimental Section). An AlGaAs tunneling barrier is grown 5 nm above the QDM layer to inhibit hole tunneling escape from the QDM, while electrons can freely escape at a rate determined by V G . We have previously used similar approaches to achieve selective optical charging of single QDs with electrons or holes [28,29]. Consequently, using our approach spin state preparation and control is possible at precisely and arbitrarily adjustable coupling conditions by controlling the polarization and frequency of the optical charging laser relative to the discrete absorption resonances of the QDM. Moreover, we demonstrate that the optical charging process can be repeated to sequentially switch from the n = 0 to 1 to 2 hole state, opening the way to access S -T 0 logical qubits that are insensitive to magnetic and electric field noise to first order while being fully tunable within the two-spin logical state space. [16] II. RESULTS

A. Measurement Scheme

The sample investigated was grown by solid-source molecular beam epitaxy (MBE) and is an n-i-Schottky photometer with a 315 nm thick i-region. Two layers of vertically stacked self-assembled InAs QDs were grown at the mid-point of the i-region and have a wetting layer-towetting layer spacing of s = 7.6 nm. The dot height was precisely fixed at h = 2.2 nm using the In-flush method [30]. After growth of the top QD layer, a 10 nm thick GaAs capping layer was deposited before a 20 nm thick Al x Ga (1-x) As tunnel barrier was grown (x = 0.33). As depicted schematically in Figure 1b, this barrier serves to prolong the tunneling time of the hole compared to the electron, thereby allowing selective optical charging [28]. Since both n-contact and the surface metallic electrode are ≥ 100 nm away from the QDM, tunneling induced charging from the contacts into the QDM is inhibited. Thereby, the QD molecule in our sample is largely decoupled and the optically prepared S T + S B charge state remains unaffected by tunneling from the contacts for the bias conditions used during operation.

Our measurement scheme for charging and probing a two hole state is illustrated schematically in Figure 1a. It consists of four phases: Reset (I), charging of first hole (II), charging of second hole (III) and readout (IV).

The four phases are schematically depicted in Figure 1b. During phase I, a strong reverse bias of V G = V I = -4V is applied for 550 ns. V G induces an axial electric field F = (V G -V BI )/d I along the growth direction, where V BI is the built-in voltage and d I is the thickness of the intrinsic region. A strong electric field, as applied in phase I, facilitates fast tunneling escape of both electrons and holes from the molecule to initialize the QDM into the neutral state, with n = 0 holes. In phase II of the measurement protocol, the gate voltage is tuned to V II in the range ≤ -0.8 V to produce electric field conditions for which the photogenerated electron tunnels out of the molecule on timescales faster than the neutral exciton lifetime, while the hole remains stored [28]. Phase II lasts 400 ns during which a 200 ns laser pulse tuned resonant to the neutral exciton transition (X 0 ) in the bottom QD is gated on using an acousto-optical modulator (AOM -red colored pulse in Figure 1a). The resonant nature of the excitation combined with the discrete electronic structure of the QDM ensures that the n = 1 charge state is reached. Hereby, a maximum of one single electron-hole pair is generated in the QDM and, thereby, the dot remains optically active until charged by a single hole, whereupon the absorption of the trion shifts out of resonance with the driving laser field. In this way, the resonant excitation ensures that only a single hole is generated. Once the n = 1 hole charge state has been reached, the discrete absorption of the QDM shifts to one of the positively charged trion transitions. As such, the molecule becomes photosensitive again by tuning the driving laser frequency or switching V G to induce a DC Stark shift and re-establishing resonance with the positive trion (X + ). Thus, as depicted schematically in Figure 1b (III), moving from the n = 1 to n = 2 charge configuration involves switching the laser frequency or electric field to a new value (V G = V III ) whereby X + is resonantly excited. As before, a photon is absorbed whereupon the photogenerated electron tunnels out of the QDM leaving two holes in the system. Once the QDM has been optically charged with two holes, the voltage is switched to a higher level (V IV ≥ -0.6 V), for which electrons no longer tunnel out of the molecule and quantum optical experiments such as luminescence or resonance fluorescence can be performed to confirm the presence of the optically generated hole(s) in the QDM. Figure 1b (IV) depicts the readout of the charge state, denoted phase IV. Readout of the charge status n in the QDM is performed by tuning a third laser field into resonance with an excited state transition of either X 0 , X + or the doubly charged positive trion (X ++ ) to pump a luminescence recycling transition. Due to the quantum confined stark effect (QCSE) [31], the laser energies used to charge the QDM are significantly frequency detuned by ∆ = -1470 GHz ( δ ∼10 kHz linewidth), from the readout laser allowing clean spectral filtering between charging and readout signals. As soon as two holes are prepared inside the QDM, the gate voltage is widely adjustable during the readout phase IV, e.g. to control spin-spin orbital coupling in the 2h-molecule and explore the S -T 0 qubit state space. The boundaries are imposed by the voltage where electron tunneling is fast compared to the measurement time in reverse bias (-0.6 V) and the diode is flooded with carriers in forward bias (0.7 V). Within this range, any coupling condition can be addressed since the charge state is pre-set. For the measurements presented here the scheme presented in Figure 1a is continuously repeated at 420 kHz.

Depending on the charge state required we switch between n = 0, 1 and 2 hole charging. This is done by either gating off the charging laser or tuning the gate voltage such that the laser does not match the resonance condition with either the X 0 transition (1h) or the X 0 and X + transitions (2h). To identify readout transitions for probing zero, one, and two hole states, we recorded voltage dependent photoluminescence data. Figure 1c shows typical voltage dependent photoluminescence recorded under non-resonant excitation into the wetting-layers (1458 meV) as a function of the DC voltage (V IV ) applied to the Schottky contact. For this measurement the reset phase I is applied, while the charging pulses in phases II and III of the measurement scheme are gated off. By exciting electron hole pairs in the wetting layers, charging of the QDM occurs probabilistically. Thus, the charge occupancy statistically fluctuates leading to photoluminescence signals from different charge states in the time integrated spectrum that allows simultaneous monitoring of different charge states. Crossings and avoided crossings characteristic for QDMs are observed. They arise due to the orbital hybridisation of hole states. Hybridisation takes place in both, ground and excited state, leading to charge state specific patterns [32,33]. In this way X 0 , X + , and X ++ transitions are identified and marked in Figure 1c. These transitions link the n=0, 1 and 2 hole ground to excited states and are therefore used to probe the resulting charge state of the QDM after switching on the 1h and / or 2h charging laser pulses.

B. One Hole Charging

To evaluate the performance of the all-optical 1h charging scheme outlined above, we implemented the experimental protocol depicted in Figure 1a including only the 1h charging pulse (II), while omitting the 2h charging pulse (III). During the readout phase IV of the measurement the QDM was excited via a luminescence cycling p-p transition. In this, as well as in the following experiment, we drive transitions where the excited state electron is located in the lower quantum dot. As depicted in Figure 2a, the neutral exciton p-p transition (X 0 [p-p] ) and the positive trion p-p transition (X + [p-p] ) are energetically overlapping. This provides a cycling transition for both, the vacuum and the 1h ground state. Even if the excitation is not charge state selective, the resulting emission is. The X 0 / X + emission obtained when exciting the vacuum/ 1h ground state are separated spectrally. Therefore, we can deduce from the emission intensities on the predominant charge ground state.

The data presented in Figure 2b specifically compare measurements performed with (blue) and without (red) resonant 1h charging pulse applied. V IV was set to -0.2 V. If the 1h charging laser is blocked then the X 0 emission is observed, since the QDM is remains uncharged. In contrast, upon gating the charging laser on during phase II of our measurement cycle, the X 0 emission signal vanishes during the readout phase of the measurement. This reflects the fact that, if charging has occurred the discrete excited state of the charge neutral exciton X 0 [p-p] is no longer optically active. On the other hand, when the 1h charging pulse is blocked, X + emission is not observed whereas gating on the 1h charging laser results in an anticorrelated increase in the intensity of X + emission at the expense of X 0 emission. These observations clearly indicate that selective, all optical single hole charging of the molecule has taken place.

To identify X 0 resonances for charging one hole, the integrated intensity of X 0 and X + were monitored as a function of the 1h charging laser frequency (ω 1h ), with the device biased at the charging voltage V II . Figure 2c shows a 2D false-color map of the integrated emission intensity from X 0 for varying V II and ω 1h (left). The white dashed line marks V II = -1.1 V, where a line cut of X 0 and X + emission along energies between 1315 meV and 1321.5 meV is recorded and presented in the rightmost panel. When the 1h-charging laser matches the energy of a transition related to X 0 , an electron hole pair is generated and single hole charging occurs. A transition related to X 0 can include s-orbital and excited state X 0 transitions of both dots forming the molecule, as well as indirect exciton transitions. Tunnel ionization leaves one hole in the ground state resulting in a weakening of the X 0 emission observed during the readout phase of the measurement. This occurs at 1318 meV and 1320 meV as shown in the line cut presented in Figure 2c. Concurrently, an anti-correlated increase of the X + emission is expected and observed as a fingerprint of the deterministic single hole charging. Thus, monitoring X 0 and X + emission intensities allows identification of 1h charging transitions in a voltage regime where charging takes place.

Besides adjusting the frequency of the charging laser the optimal charging voltage has to be identified. For increasingly negative V G the axial electric field becomes larger and hole tunneling times become shorter than the temporal width of the charging plateau (phase II). A similar statement applies to the electron tunneling times becoming too long as V G becomes more positive and the axial electric field reduces. For both cases, the probability of selective charging during phase II of our measurement protocol reduces. A compromise between efficient electron tunneling and sufficiently long hole retention times is found by sweeping ω 1h for different V II (Figure 2c, left). The detected resonance energies reduce with decreasing V II due to the DC Stark effect. In addition, as the electron tunneling becomes faster the lifetime of the excited state decreases, broadening the linewidth of the charging resonance. By analyzing the voltage-dependent full-width half-maximum of the resonance at V II , we estimate the electron tunneling time to be ≤ 2 ps [34].

Based on the data presented in Figure 2 and the above discussion, we select V II = -1.08 V while the charging laser energy is fixed at ω 1h = 1317.8 meV to generate a single hole in the molecule. This optimal working point is marked by a red cross in Figure 2c and are used in the next section for sequential 2h charging of the QDmolecule.

C. Two Hole Charging

As discussed above in relation to Figure 1, sequential optical charging of the QDM from n = 0 to 1 and 2 is achieved by switching on the charging laser during phase III of the measurement protocol. Analogously to the 1h charging experiment discussed in the previous section, we repeated the optimization procedure to find V III and ω 2h , while using the optimal parameters V II and ω 1h found for 1h charging. The green and red curves presented in the line cut of Figure 3a show the integrated emission intensity of X 0 as a function of the laser energy in phase III of our protocol ( ω 2h ), with and without the 1h charging laser having previously been applied in phase II. V III is set to match V II , making the charging resonances of both phases comparable. Since X + and X ++ optical transitions partly spectrally overlap, the two hole charging resonances are in a first step identified indirectly via the reduction of the X 0 transition intensity. Reducing the laser power of both charging pulses to 60 % of saturation (240 nW) allows to monitor the X 0 emission, even if charging takes place. The red curve shows resonances (e.g. at 1317.8 meV) observed when applying only the second charging pulse. As in the data presented in Fig. 2c, 1h is charged by the excitation of X 0 , leading to the observation of the same resonances. In contrast, when previously applying the first charging pulse (green curve), the second pulse can only excite the positive trion. As the X + energy is shifted compared to the X 0 energy, we observe a shift of resonances (e.g. at 1318 meV). The X + resonance is then used to charge a second hole into the QDM.

Similar to situation discussed in the context of Fig. 2c, the DC Stark shift of the X + transition is observed when recording the intensity of the X 0 emission while varying V III and ω 2h . Typical results are presented in Figure 3a. The energy and voltage of phase II are chosen as described above. A comparison with Figure 2c helps to identify voltages for which sequential 1h and 2h hole charging is achievable for the same resonant laser energy. This allows to sequentially charge two holes into the QDM with only one laser energy by modifying the gate voltage while the resonant charging laser is gated on. Utilizing one laser only, as done in the following experiments is desirable to reduce the complexity of the charging scheme. When maintaining the 1h charging energy ω 1h = 1317.8 meV, the voltage of phase III has to be set to V III = -1.34 V to enable the sequential charging of a second hole during phase III. The conditions to have sequential resonances between the charging laser and the 1h and 2h charging voltages are indicated in Figure 3a by red and green crosses, respectively.

Up to now, the charging of a second hole has been demonstrated only indirectly via the reduction of the X 0 emission. In the following, the presence of two holes in the QDM is directly verified by performing resonance fluorescence on the doubly charged exciton transition X ++ . In contrast to the measurements discussed above, where readout was performed via luminescence by pumping an excited state of the neutral (X 0 [p-p] ) or positively charged exciton (X + [p-p] ), here we resonantly excite and probe the s-shell doubly charged exciton X ++ transition during phase IV of our measurement. To filter out the readout laser, a cross-polarized resonance fluorescence setup was used [35]. Figure 3b shows the X ++ emission upon charging zero, one, and two holes into the QDM in phases II and III of the measurement. For all three datasets the charging laser remained on throughout phases II and III of the measurement. V G was adjusted to facilitate either 0h, 1h, or 2h charging. When charging zero and one hole(s) the signal consists mainly of unsuppressed laser. However, as soon as the QDM is loaded with two holes, the X ++ emission rises by a factor of > 18. Besides the main emission peak at 1322.95 meV, a side peak at 1323.05 meV is observable. This side peak is identified as an indirect X ++ transition by comparison with Fig- ure 1c. The observed increase of the emission intensity proves that 2h selective optical charging of the QDM has occurred.

Even if 1h and 2h charging is demonstrated, the experiments presented do not conclusively show that holes are sequentially added to the QDM, i.e. that the charge status is sequentially shifted from n = 0 to 1, to 2 using optical pumping. To prove the sequentiality of the charging in our experiments and determine the typical photon fluxes required for charging, we continue to show that charging takes place primarily during the intended phases of the voltage sequence (II and III) and, thereby, confirm the sequential, all optical preparation of the 2h Comparison of X 0 emission with (green curve) and without (red curve) 1h charging laser applied, when sweeping phase III laser energy. The charging resonances shift as the system switches between resonantly addressing the X 0 and X + transition for charging on and two holes, respectively. (b) Resonance fluorescence emission of the X ++ transition for charging zero / one/ two holes (blue/ red/ green).

charge state of the QDM.

D. Sequential Charging

To confirm sequentiality of the charging process, we performed a measurement using a single charging pulse having a temporal width of 400 ns. This charging pulse was swept through the two charging plateaus of phase II and III, each of which is 200 ns long. This procedure is illustrated schematically in Figure 4, showing the measurement sequence and the temporal advance of the charging laser pulse τ . At τ < 0, the charging laser overlaps solely with phase I of the measurement sequence and is, therefore, ineffective in charging the QDM. The charging pulse is then shifted through phases II (1h charging), III (2h-charging) of the measurement protocol until it overlaps solely with phase IV (readout). By advancing the time when the charging pulse is applied, its temporal overlap with the two charge plateaus changes. Figure 4 shows the emission of X 0 , X + and X ++ transitions as a function of the temporal advance τ and the applied charging power. The charging energy and the voltage plateaus of phase II and III are set as previously defined for two hole charging. In addition, the second charging laser was turned off for this experiment. Readout was performed via luminescence driven by pumping p-shell transitions at -0.2 V to detect X 0 (Figure 4 -upper panel), X + (middle panel) and X ++ (lower panel) emission simultaneously.

Complete embedding of the charging pulse in phase I (τ < 0 ns) does not result in charging. X 0 therefore dominates the X + and X ++ emission. At τ = 0 ns, the laser pulse enters the 1h charging plateau of phase II. As a result, the emission intensity of X + rises as charging of the QDM with a single hole takes place. At the same time, the emission intensity of X 0 reduces while the X ++ remains close to the background level indicative of the charge state of the QDM being enhanced to n = 1. For low power, this effect occurs at longer values of τ , which corresponds to a larger overlap between laser pulse and charging plateau. This reflects the fact that the charging efficiency is reduced -charging is probabilistic due to the tunneling process and weaker photon fluxes require longer times to establish the n = 1 charge state. The two hole charging plateau is reached at τ = 200 ns. While the X 0 emission remains at a low level, the intensity of X ++ progressively rises with τ and the intensity of X + simultaneously reduces in an anti-correlated manner. This key observation shows that the number of charged holes sequentially increased from n = 1 to 2. The second hole is accordingly mainly charged during phase III due to the finite orbital degeneracy of the states excited. Up to τ = 600 ns 2h charging is performed, as the laser pulse overlaps with both charging plateaus. After τ = 600 ns the pulse leaves the first charging plateau, whereupon n = 1 charging has not occurred anymore. As a result, the X + transition cannot be excited during phase III, and the charge status of the QDM remains close to n = 0. Consequently, the emission from X 0 reappears while X ++ progressively decreases. We attribute the small increase of X + emission with τ to the nonzero probability of charging one hole during phase III. As the resonances broaden with decreasing voltage and X 0 and X + charging transitions spectrally overlap, the selectivity of the charging process is reduced. This makes generation of unwanted charge states more likely. At τ = 0 ns, the charging laser enters the first charging plateau. The readout is performed via p-shell excitation while detecting the spectrally detuned s-shell emissions of X 0 , X + and X ++ . Below: Integrated emission of X 0 , X + and X ++ (top to bottom) is logarithmically plotted for varying charging pulse delay and charging power. Magenta dashed lines indicate the edges of the 1h and 2h charging plateaus.

III. DISCUSSION AND SUMMARY

To quantify the efficiency of the proposed charging scheme we calculated P(n|m), the conditional probability for charging n holes, given m charging pulses are applied. The reset probability is extremely close to unity (P(0|0) = 1), and considering only states with n ≤ 2 to reflect the degeneracy of a single QD orbital, we obtain 1h/ 2h charging probabilities of ≥ 93.5±0.8 %/ ≥ 80.5±1.3 %, respectively (see section 4.2). These results show that the sequential, all-optical preparation of desired number of charges is a robust and reliable process. Certainly, the optical preparation of charges is more complex compared to QDM charging via tunneling from a diode contact. Additional voltage plateaus and laser pulses are required. Furthermore, the repetition rate of the sequence is currently limited to 740 kHz due to the duration of reset and charging pulses. However, this limitation is mainly caused by the RC time constant of the diode. By using micron scale photodiodes having low RC time constants [36], this can be increased to up to 500 MHz. Moreover, it is superior to optimize protocols for increasing the readout phase IV duty cycle. This can be done by performing several readout cycles after having previously prepared the charge state. The limit of such approaches will be determined by the extent to which measurement back action influences the charge status of the QDM, e.g. via Auger auto-ionization [37]. Furthermore, the demonstrated scheme does not pose any restrictions on the readout voltage to obtain certain charge states. The X 0 , X + and X ++ intensities shown in Figure 4 are acquired at the same gate voltage, showing its independence.

In summary, we have proposed and demonstrated a four-phase measurement sequence that allows independent control of charge status and inter-dot coupling of a single, electrically tunable QDM. We demonstrated one and two hole charging via optical excitation and tunnel ionization. The addressability of the generated charge state was thereafter not affected by the gate voltage, facilitating electrically-tunable spin-spin interactions induced using the exchange coupling between the two spins [38]. By combining efficient charging and separate gate voltage control the proposed method of optical charging is therefore suitable to replace and outperform previous charging approaches with respect to controllability and flexibility. Independent charge state preparation paired with the ability to manipulate the inter-dot coupling paves the way for protocols requiring simultaneous spin and coupling control, as for example needed for 2D cluster state generation. [18] IV. EXPERIMENTAL SECTION

A. Sample Structure

The QDM investigated was grown by solid-source molecular beam epitaxy. It consisted of two laterally stacked InAs quantum dots embedded in a GaAs matrix. The inter-dot coupling strength is determined by the wetting layer to wetting layer separation of 7.6 nm. The individual dot height is fixed to 2.2 nm via an Inflush technique. Thereby, the energies of the two QDs are adjusted. The upper QD is designed to have a higher confinement energy compared to the lower QD to facilitate electric field induced tunnel coupling of orbital states in the valence band [39]. A 20 nm thick Al x Ga (1-x) As tunnel barrier (x = 0.33) was grown 10 nm above the QDM to prolong hole tunnelling times and enable tunnel ionization charge state preparation. Furthermore, the molecule is embedded into an n-i-Schottky diode to apply electric fields along the growth direction of the sample. The n-doped region and the Schottky surface metallic electrode are used as contacts. Both diode contacts are more than 100 nm away from the molecule to prevent uncontrolled charge tunnelling into the QDM and, nonoptical or selective modification of the charge status. All measurements are performed at 10 K. For preparation and readout of the charge state tunable diode lasers are used.

B. Fidelity Calculation

To estimate a lower boundary of the one and two hole charging fidelities, we assume that the reset phase I is perfect: P(0|0) = 1. This expectation is likely to be achived in our experiment since the discharging electric field is high and the duration of the initialization phase of the measurement is sufficiently long. Here, P(n|m) is the probability of charging n holes, given m charging pulses are applied. This assumption can be justified, as duration and voltage of phase I can be chosen to make the presence of charges negligible. Furthermore, only charge states with n ≤ 2 are taken into account due to the orbital structure of the charged QDM.

By monitoring the X 0 emission with and without applying one charging pulse, we can calculate the probability of charging a non-zero number of holes P(¬ 0|1) =

N 0 0 -N 0 1 N 0 0 .
Where N a b is the number of counts observed for transition X a , with a ∈ {0, +, ++}. b ∈ {0, 1, 2} on the other hand indicates the number of charge pulses applied. The probability for selectively charging one hole, given that we want to charge a single hole, can be written as P(1|1) = P(¬ 0|1) -P(2|1).

The probability for charging two holes when only one is intended can be written as: P(2|1) ≤

N ++ 1 N ++ 2
. We normalize the counts obtained monitoring the X ++ transition when planned to charge one hole by the counts of the X ++ transition when planned to charge two holes. This indicates in which proportion of the overall cases we charge the dot with two holes if we only want to have one. In the case of a perfect two hole charging procedure, this inequality changes to an equation.

The fidelity of charging one hole can now be written as:

P (1|1) ≥ N 0 0 -N 0 1 N 0 0 - N ++ 1 N ++ 2 (1)
The two hole charging fidelity can be calculated by estimating its counter events: P(2|2) = 1 -P(1|2) -P(0|2). The probability of charging zero holes for this case is estimated via the counter event of charging zero holes, given a charging pulse is applied: P(0| 2) ≤ 1 -P(¬ 0|1). The probability for loading one hole given two holes are intended, we follow the same argumentation as for calculating P(2|1). However, we monitor the X + transition and divide the counts of the two hole by the one hole charging case: P(1|2) ≤

N + 2 N + 1 .
This allows us to write the two hole charging fidelity as:

P (2|2) ≥ N 0 0 -N 0 1 N 0 0 - N + 2 N + 1 (2)
V. ACKNOWLEDGEMENTS

We gratefully acknowledge financial support from the German Federal Ministry of Education and Research (BMBF) via Q.Link.X (16KIS0874), QR.X (16KISQ027), the European Union's Horizon 2020 research and innovation program under Grant Agreement 862035 (QLUSTER) and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via SQAM (FI947-5-1), DIP (FI947-6-1), and the Excellence Cluster MCQST (EXC-2111, 390814868). FB gratefully acknowledges the Exploring Quantum Matter (ExQM) programme funded by the State of Bavaria. F.S., K.B., and K.M. gratefully acknowledges the BMBF for financial support via project MOQUA (13N14846).

TimeFIG. 1 .

 1 FIG. 1. Optical charging of QDM. (a) Measurement scheme for charging and probing two hole states, consisting of four phases: I Reset, II Charging one hole (1h), III Charging two holes (2h), IV Readout. The black line symbolizes the gate voltage VG with voltage plateaus VI to VIV . The colored boxes indicate laser pulses for charging and readout. (b) Band structure of QDM with double potential well (red, blue) and tunneling barrier (grey). Four sequence phases are illustrated: I -Reset: Low gate voltage leads to tunneling of electrons and holes to empty QDM, II, III -Charging one/ two holes: Resonant laser pulse creates electron hole pair. Via tunneling ionization, charges are separated and QDM loaded with one/ two holes, IV Readout: Resonant s-or p-shell excitation is applied to probe charge state. (c) Voltage dependent photoluminescence measurement showing X 0 , X + and X ++ transitions.

FIG. 2 .

 2 FIG. 2. Emission of X 0 and X + when charging one hole. (a) Excitation scheme used to identify 1h charging resonance. Due to overlapping p-shells, X 0 (blue)/ X + (red) is detected when vacuum (vac)/ 1h ground state is predominant. (b) Emission spectrum at -0.2 V under p-shell excitation with and without charging pulse of phase II applied (red/ blue). X 0 transition fades when charging takes places, while X + rises. (c) 2D colormap showing the X 0 intensity as a function of VII and ω 1h . Yellow (blue) indicate high (low) count rates during the readout phase IV of the measurement. The red cross indicates the settings used in the rest of the manuscript for 1h charging. White dashed line marks -1.1 V, from where the line cut is extracted (right). Example of the anti-correlated integrated intensities of X 0 and X + transitions when modifying the energy of the 1h charging laser to identify charging resonances at phase-II of the measurement (right).

FIG. 3 .

 3 FIG. 3. Two hole charging resonances. (a) VIII and ω 2h dependent X 0 emission (left), showing shifted resonances for 2h charging, compared to Figure 2c. The red (green) cross indicates the voltage used for one (two) hole charging. The red dashed line illustrates the required charging energy. White dashed line marks VIII = -1.08 V at which a line cut is presented (right).Comparison of X 0 emission with (green curve) and without (red curve) 1h charging laser applied, when sweeping phase III laser energy. The charging resonances shift as the system switches between resonantly addressing the X 0 and X + transition for charging on and two holes, respectively. (b) Resonance fluorescence emission of the X ++ transition for charging zero / one/ two holes (blue/ red/ green).

FIG. 4 .

 4 FIG. 4.Sequential two hole charging. Measurement sequence, visualizing the temporal sweep (τ ) of the 400 ns charging pulse over the two 200 ns charging voltage plateaus. At τ = 0 ns, the charging laser enters the first charging plateau. The readout is performed via p-shell excitation while detecting the spectrally detuned s-shell emissions of X 0 , X + and X ++ . Below: Integrated emission of X 0 , X + and X ++ (top to bottom) is logarithmically plotted for varying charging pulse delay and charging power. Magenta dashed lines indicate the edges of the 1h and 2h charging plateaus.

Key words: Quantum dot molecule, charge state control, inter-dot coupling, charge storage, optical charging


Pierre Monnin
email: pierre.monnin@orange.com

Adrien Coulet
email: adrien.coulet@inria.fr

Matching Pharmacogenomic Knowledge: Particularities, Results, and Perspectives ⋆

Keywords: Instance Matching, n-ary Tuple, Preorder, Graph Embedding, Pharmacogenomics

Knowledge in pharmacogenomics (PGx) is scattered across several resources, e.g., reference databases and the biomedical literature. Matching their content would thus lead to a consolidated view of the available PGx knowledge that could, in turn, support multiple downstream applications, including knowledge curation and precision medicine. However, matching atomic units of PGx knowledge is challenging due to their peculiarities: they are of n-ary nature, represented with heterogeneous vocabularies, and with various levels of granularity. In this paper, we frame the matching of PGx knowledge units of various provenance as an instance matching problem. We summarize our work to represent such units within a knowledge graph named PGxLOD, and to match them with a rule-based and a graph embedding-based matching approaches. We then particularly discuss the remaining challenges and how our research artifacts opened to the community could foster new benchmarks and methods for structure-based instance matching.

Introduction

The increasing adoption of Linked Open Data (LOD) principles as well as Semantic Web standards and technologies leads to an ever-growing number of resources being published online. Consequently, the knowledge of a domain can be scattered across several complementary, potentially overlapping, resources. That is to say, both similar and complementary knowledge units may be represented across different resources. The aforementioned standards and technologies also allow the concurrent edition of resources by human and software agents. This can lead to duplicates within the same resource. Thus, matching the content of such resources is a first and necessary step towards offering a consolidated view ⋆ This work was supported by the French National Research Agency (ANR) through the PractiKPharma project (ANR-15-CE23-0028 -http://practikpharma.loria.fr).

of the available knowledge of a domain. However, matching similar knowledge units within and across resources requires to face issues such as difference of vocabularies or levels of granularity in the representation of knowledge units. This general observation is also valid in pharmacogenomics (PGx), a domain that studies the influence of genetic factors on drug response phenotypes. Indeed, state-of-the-art knowledge in PGx is mainly scattered across specialized databases such as PharmGKB, and the biomedical literature, i.e., PubMed. Electronic Health Records (EHRs) can also be mined to extract PGx knowledge. Additionally to this scattering, PGx knowledge also suffers from heterogeneous levels of validation. Indeed, some PGx knowledge units have been extensively studied, validated, and are implemented in clinical practice. On the contrary, others have only been observed on reduced cohorts of patients and remain to be further studied and confirmed. Consequently, matching PGx knowledge across resources would, first, offer a consolidated view of the available knowledge of the domain; and second, should ease knowledge curation. Indeed, PharmGKB is manually fed by human curators who continuously review the literature. Connecting similar PGx knowledge units across articles would ease their work by guiding them to sets of relevant articles to consider jointly and confront. This would facilitate the validation, or moderation, of state-of-the-art knowledge.

In this paper, we specify the problem of matching PGx knowledge units of various provenance and its challenges (Section 2). Then, we summarize our research results consisting of a knowledge graph (KG) named PGxLOD, and two matching approaches (Section 3). Finally, we outline new research directions, and advocate for the community reuse of our produced research artifacts (Section 4).

Problem Setting

Matching PGx knowledge requires to tackle several specific issues. Indeed, PGx knowledge units are n-ary relationships whose arguments are sets of individuals, e.g., the sets of involved drugs, the sets of involved genetic factors, and the sets of involved phenotypes (Fig. 1a). Such a relationship states that a patient being treated with the specified drugs while having the specified genetic factors will likely experiment the specified phenotypes. To illustrate, Fig. 1b depicts a stateof-the-art PGx relationship. It should be noted that, on the application side, such knowledge units are named pharmacogenomic relationships. However, in a mathematical formalism or Semantic Web standards, such units are actually relation instances or relation tuples. To avoid any ambiguity, we will only refer to PGx knowledge units as PGx tuples.

We choose to represent PGx tuples in a KG that follows Semantic Web formalisms to interconnect with ontologies and additional LOD about drugs, genetic factors, and phenotypes. Such interconnections provide additional knowledge that we leverage during the matching process. In such formalisms, only binary predicates exist. Consequently, PGx tuples are reified: tuples become individuals that are linked to their components with binary predicates. For example, in Fig. 1c, the tuple pgt 1 is an individual linked to its components with the causes predicate. In such a view, the matching of PGx tuples comes down to an instance matching task. Due to the reification of tuples, this process will solely rely on a comparison of neighbors of tuples in the graph, which corresponds to a structure-based matching approach. Beside their arity, matching PGx tuples requires to face their incomplete and heterogeneous representations (Fig. 2). These issues inherently lead to various types of alignments between tuples, which is somehow unusual in instance matching. For example, tuples can be identical, e.g., tuples on the bottom left. Identical alignments may need to rely on translations and synonyms. Some tuples may be more specific than others, e.g., tuples on the right. Matching such granularity differences may rely on domain-specific orderings such as ontology hierarchies. In addition, a specified argument is more specific than an unknown argument, e.g., vascular disorders is more specific than ??. Tuples may also be related when their arguments are comparable w.r.t. some orderings but not all ordered in the same way (e.g., tuples at the bottom). This type of alignments can also connect tuples whose arguments are somehow related without being strictly comparable. Example of heterogeneity issues and expected matching results (dashed gray arrows) between PGx tuples. The phenotype is unknown (denoted ??) for one tuple, coumaphène and hémorragie are the French words for warfarin and hemorrhage, which in turn is a synonym of bleeding. CYP2C9*2 is a gene variant that is more specific than the gene CYP2C9 itself, bleeding is more specific than vascular disorder according to an ontology, and vascular disorders is related somehow to warfarin adverse reaction.

To align PGx tuples represented in PGxLOD, we first proposed a symbolic rule-based approach, named tcn3r5 [START_REF] Monnin | Knowledge-based matching of n-ary tuples[END_REF]. In this approach, we see PGx tuples in their mathematical form of n-ary tuples, where each argument is a set of individuals, e.g., from Fig. 1c, pgt 1 = ({warfarin} , {CYP2C9} , {bleeding}). In this view, matching two tuples comes down to comparing their arguments pairwise, before concluding on the relatedness of the two tuples. To this aim, we define two preorders ≼ p and ≼ O that consider ontology statements, and thus enrich the comparison provided by set operators (i.e., =, ⊆). We propose in [START_REF] Monnin | Knowledge-based matching of n-ary tuples[END_REF] five matching rules that conclude on five relatedness levels between tuples. The many resulting alignments illustrate potentialities of our approach and provide insights on PGxLOD. Rule 5 that concludes on the weakest relatedness level generated the most inter-resource alignments, which emphasizes the importance of weaker relatedness levels to align resources and overcome their heterogeneity.

To cope with this need for flexibility, we considered KG embeddings models. Indeed, the continuous aspect of graph embeddings may provide the needed flexibility. We framed our task as a node clustering task performed on the embedding space6 [START_REF] Monnin | Discovering alignment relations with Graph Convolutional Networks: A biomedical case study[END_REF]. Consequently, we learn node embeddings using Graph Convolutional Networks and the Soft Nearest Neighbor Loss such that similar PGx tuples have low distances between their embeddings. Then, we apply a clustering algorithm on the node embeddings and consider nodes assigned to the same cluster as similar. To learn node embeddings, we constituted gold clusters that are based on the alignments output by our rule-based approach. We showed that integrating domain knowledge by adding inferences in the KG improves clustering performance. We also observed that distances in the embedding space are coherent with the "strength" of the different alignments (e.g., smaller distances for equivalences, larger for weak relations). This result corresponds somehow to a rediscovery of KG semantics in the embedding space.

Discussion & Perspectives

PGxLOD only contains PGx tuples from the state of the art. Hence, one major perspective resides in developing an "observational" version with results of EHR mining or EHRs themselves. However, such an integration raises several issues related to text mining and data privacy (e.g., anonymization, access control). Beside biomedical research, we think PGxLOD is a useful resource for Computer Science researchers. Indeed, we illustrated with our matching approaches the various and challenging characteristics of PGxLOD: integration of several data sets (i.e., owl:sameAs links) and ontologies (i.e., hierarchies of classes and predicates, predicate inverses and symmetry), and a medium size (i.e., scalability issues). For these reasons, PGxLOD constitutes an interesting real-world KG to experiment matching approaches. That is why, we envision to propose its consideration in the Ontology Alignment Evaluation Initiative.

Both our matching approaches showed how domain knowledge and reasoning mechanisms can serve a structure-based matching. These approaches also present complementary strengths and could foster each other. Two perspectives now lie in (i) learning new rules from clusters output by our embedding-based approach and (ii) updating alignments between PGx tuples when new tuples are integrated. Other aspects of PGx knowledge and metadata are not currently considered but pave the way for future works. For example, PGxO allows to represent negation within PGx tuples (e.g., drugs not causing a tuple). Consequently, we could match contradictory tuples, which raises several questions such as the geometric representation of contradiction in the embedding space. We could also tackle the task of knowledge validation, i.e., confirming or moderating a knowledge unit based on similar or contradictory units existing in other resources. This would require to leverage alignments and heterogeneous quality metadata such as levels of evidence in PharmGKB, or odd ratios in biomedical articles. Such a knowledge validation approach would, in turn, realize our ultimate objective of offering a consolidated view of PGx knowledge to clinicians.

Fig. 1 .

 1 Fig.1. Graphical representation of an abstract (1a), an example (1b), and a reified (1c) pharmacogenomic relationships. The example relationship states that patients having the "*4" version of the CY P 2D6 gene will not experience the expected effect of codeine. gf stands for genetic factor, d for drug and p for phenotype.

  Fig.2. Example of heterogeneity issues and expected matching results (dashed gray arrows) between PGx tuples. The phenotype is unknown (denoted ??) for one tuple, coumaphène and hémorragie are the French words for warfarin and hemorrhage, which in turn is a synonym of bleeding. CYP2C9*2 is a gene variant that is more specific than the gene CYP2C9 itself, bleeding is more specific than vascular disorder according to an ontology, and vascular disorders is related somehow to warfarin adverse reaction.

https://pgxo.loria.fr -https://pgxlod.loria.fr

https://github.com/pmonnin/tcn3r

https://github.com/pmonnin/gcn-matching

Representing and Matching PGx Knowledge

We developed our own ontology, PGxO, and our own KG, PGxLOD 4 [1], to represent PGx tuples of various provenance. PGxO provides a reduced set of classes and predicates to represent reified PGx tuples. We populated PGxO to create PGxLOD by extracting 50,425 PGx tuples from three main resources: structured data of PharmGKB (3,650 tuples), semi-structured data (namely clinical annotations) of PharmGKB (10,240 tuples), and PubMed abstracts (36,535 tuples). PGxLOD also includes knowledge about drugs, genetic factors, and phenotypes by integrating several LOD graphs and ontologies (e.g., DisGeNET, DrugBank, MeSH). PGxLOD is publicly available and respects LOD and FAIR principles.


Tsung-Hsuan Tsai
email: tsai71517@gmail.com

Phase transition for the existence of van Kampen 2-complexes in random groups

Gromov showed that [Gro93] with high probability, every bounded and reduced van Kampen diagram D of a random group at density d satisfies the isoperimetric inequality |∂D| ≥ (1 -2d -s)|D|ℓ. In this article, we adapt Gruber-Mackay's prove [GM18] for random triangular groups, showing a non-reduced 2-complex version of this inequality.

Moreover, for any 2-complex Y of a given geometric form, we exhibit a phase transition: we give explicitly a critical density dc depending only on Y such that, in a random group at density d, if d < dc then there is no reduced van Kampen 2-complex of the form Y ; while if d > dc then there exists reduced van Kampen 2-complexes of the form Y .

As an application, we show a phase transition for the C(p) small-cancellation condition: for a random group at density d, if d < 1/(p + 1) then it satisfies C(p); while if d > 1/(p + 1) then it does not satisfy C(p).

1 Introduction Random groups. The first mention of random group presentations is the density model by M. Gromov in [Gro93, 9.B]. Formally, a random group is a random variable with values in a given set of groups, often constructed by group presentations with a fixed set of generators and a random set of relators. The goal is to study the asymptotic behaviors of a sequence of random groups (G ℓ ) when the maximal relator lengths ℓ goes to infinity. We say that G ℓ satisfies some property Q ℓ asymptotically almost surely (a.a.s.) if the probability that G ℓ satisfies Q ℓ converges to 1 as ℓ goes to infinity.

Let us consider the permutation invariant density model of random groups introduced by Gromov in [START_REF] Gromov | Asymptotic invariants of infinite groups[END_REF]p. 272] and developed in [START_REF] Tsai | Density of random subsets and applications to group theory[END_REF]. Fix the set of generators X m = {x 1 , . . . , x m } with m ≥ 2 for group presentations. Let B ℓ be the set of cyclically reduced words of X ± m of length at most ℓ. We shall construct random groups by densable and permutation invariant random subsets of B ℓ . The sequence (R ℓ ) is called permutation invariant if R ℓ is a permutation measureinvariant random subset of B ℓ .

Many natural models of random subsets are densable and permutation invariant. For example, the uniform distribution on all subsets of cardinality ⌊B ℓ ⌋ d considered in [START_REF] Ollivier | Sharp phase transition theorems for hyperbolicity of random groups[END_REF], [START_REF] Ollivier | A January 2005 invitation to random groups[END_REF] and [START_REF] Ollivier | Some small cancellation properties of random groups[END_REF], or the Bernoulli sampling of parameter |B ℓ | d-1 considered in [AŁ Ś15] for random triangular groups. where (R ℓ ) is a densable sequence of permutation invariant random subsets of (B ℓ ) with density d.

For detailed surveys on random groups, we refer the reader to [START_REF] Ghys | Groupes aléatoires (d'après Misha Gromov[END_REF] by E. Ghys, [START_REF] Ollivier | A January 2005 invitation to random groups[END_REF] by Y. Ollivier, [START_REF] Kapovich | On group-theoretic models of randomness and genericity[END_REF] by I. Kapovich and P. Schupp, and [START_REF] Bassino | Random presentations and random subgroups: a survey[END_REF] by F. Bassino, C. Nicaud and P. Weil.

Isoperimetric inequalities. In order to prove the hyperbolicity of a random group at density d < 1/2, Gromov showed in [START_REF] Gromov | Asymptotic invariants of infinite groups[END_REF]9.B] that a.a.s. reduced local van Kampen diagrams of G ℓ (m, d) satisfy an isoperimetric inequality depending on the density d. Ollivier's proof in [START_REF] Ollivier | Sharp phase transition theorems for hyperbolicity of random groups[END_REF] can achieve a slightly stronger 1 inequality

|D (1) | ≥ (1 -d - ε 2 )|D|ℓ.
One may expect such an inequality to hold for every reduced van Kampen 2complex Y with |Y | ≤ K. In [GM18, Section 2], D. Gruber and J. Mackay showed that in the triangular model of random groups2 , the above inequality holds for every non-reduced van Kampen 2-complexes Y with |Y | ≤ K if the reduction degree (Definition 2.1) Red(Y ) is added in the left hand side of the inequality.

However, the result fails in the regular Gromov density model: the condition |Y | ≤ K is not enough (see Remark 2.4). In Section 2 of this paper, we introduce the notion of complexity (Definition 2.2) to adapt Gruber-Mackay's inequality in the Gromov density model, establishing a non-reduced van Kampen 2-complex version of Theorem 1.3.

Theorem 1.4. Let (G ℓ (m, d)) be a sequence of random groups with m ≥ 2 generators at density d. Let ε > 0, K > 0. For any d < 1/2, a.a.s. every van Kampen 2-complex Y of complexity K of G ℓ (m, d) satisfies

|Y (1) | + Red(Y ) ≥ (1 -d -ε)|Y |ℓ.
Phase transition for the existence of van Kampen 2-complexes. We are now interested in the converse of Theorem 1.4: Given a 2-complex Y satisfying the inequality of Theorem 1.4, is it true that a.a.s. there exists a reduced van Kampen 2-complex of G ℓ (m, d) whose underlying 2-complex is Y ?

A 2-complex Y is said to be fillable by a group presentation G = X|R (or by the set of relators R) if there exists a reduced van Kampen 2-complex of G whose underlying 2-complex is Y . An edge of a 2-complex is called isolated if it is not adjacent to any face. Since isolated edges do not affect fillability, we will only consider finite 2-complexes without isolated edges in the following.

To better formulate the problem, we consider a sequence of 2-complexes (Y ℓ ) and introduce the notion of geometric form of 2-complexes (Y, λ) (Definition 3.1), together with its density dens Y and its critical density dens c Y (Definition 3.2). The main result of this article is the phase transition at density 1dens c (Y ), for the fillability of the 2-complex Y ℓ .

Theorem 1.5. Let (G ℓ (m, d)) be a sequence of random groups with m ≥ 2 generators at density d. Let (Y ℓ ) be a sequence of 2-complexes with some geometric form (Y, λ).

(i) If d < 1 -dens c Y , then a.a.s. Y ℓ is not fillable by G ℓ (m, d). (ii) If d > 1 -dens c Y and Y ℓ is fillable by B ℓ , then a.a.s. Y ℓ is fillable by G ℓ (m, d).
In Section 3, we prove Theorem 1.5 using the multidimensional intersection formula for random subsets (Theorem 3.6, [Tsa21, Theorem 3.7]), which generalizes the proof for the C ′ (λ) phase transition in [START_REF] Tsai | Density of random subsets and applications to group theory[END_REF]Theorem 1.4]. We will see in Remark 3.3 that the second assertion of the theorem is equivalent to the following corollary.

Corollary 1.6. Let (G ℓ (m, d)) be a sequence of random groups with m ≥ 2 generators at density d. Let s > 0 and K > 0. Let (Y ℓ ) be a sequence of 2-complexes of the same geometric form such that Y ℓ is fillable by

B ℓ . If every sub-2-complex Z ℓ of Y ℓ satisfies |Z (1) ℓ | ≥ (1 -d + s)|Z ℓ |ℓ, then a.a.s. Y ℓ is fillable by G ℓ (m, d).
Note that we need Y ℓ to have at least one filling by the set of all possible relators B ℓ . It is automatically satisfied for planar and simply connected 2-complexes. In addition, if every face boundary length of Y ℓ is exactly ℓ, then the given inequality is equivalent to an isoperimetric inequality similar the inequality of Theorem 1.3. Hence the following corollary.

Corollary 1.7. Let (G ℓ (m, d)) be a sequence of random groups with m ≥ 2 generators at density d. Let s > 0 and K > 0. Let (D ℓ ) be a sequence of finite planar 2-complexes of the same geometric form such that every face boundary length of

D ℓ is exactly ℓ. If every sub-2-complex D ′ ℓ of D ℓ satisfies |∂D ′ ℓ | ≥ (1 -2d + s)|D ′ ℓ |ℓ, then a.a.s. D ℓ is fillable by G ℓ (m, d).
It is mentioned in [OW11, Proposition 1.8] that when d < 1/(p + 1), a.a.s. a random group at density d has the C(p) small cancellation condition. As an application of Theorem 1.5, we show that there is a phase transition: if d > 1/(p + 1), then a.a.s. a random group at density d does not have C(p) (see Proposition 4.2).

Acknowledgements. The content of this article is completed during my PhD thesis [Tsa22, Chapter 4] at the University of Strasbourg. I would like to thank my thesis advisor, Thomas Delzant, for his guidance and interesting discussions on the subject.

Isoperimetric inequality for van Kampen 2-complexes

We shall prove Theorem 1.4 in this section.

Van Kampen 2-complexes. We consider oriented combinatorial 2-complexes and van Kampen diagrams as in [START_REF] Lyndon | Combinatorial Group Theory[END_REF].

A 2-complex is a triplet Y = (V, E, F ) where V is the set of vertices, E is the set of oriented edges and F is the set of oriented faces. Every edge e ∈ E has a starting point α(e) ∈ V , an ending point ω(e) ∈ V and an inverse edge e -1 ∈ E, satisfying α(e -1 ) = ω(e), ω(e -1 ) = α(e) and (e -1 ) -1 = e. A geometric edge is a pair of inverse edges e, denoted by e. Every face f ∈ F has a boundary ∂f , which is a cyclically reduced loop of the under lying combinatorial oriented graph Y (1) = (V, E), and an inverse face f -1 ∈ F satisfying ∂(f -1 ) = (∂f ) -1 and (f -1 ) -1 = f . A geometric face is a pair of inverse faces {f, f -1 }, denoted by f . We denote |Y (1) | the number of geometric edges and |Y | the number of geometric faces.

A van Kampen 2-complex with respect to a group presentation G = X|R is a 2-complex Y = (V, E, F ) with two compatible labeling functions: labels on edges by generators ϕ 1 : E → X ± , and labels on faces by relators ϕ

2 : F → R ± . Compatible means that (V, E, ϕ 1 ) is a labeled graph, ϕ 2 (f -1 ) = ϕ 2 (f ) -1 and ϕ 1 (∂f ) = ϕ 2 (f ).
The data of the labels ϕ 1 , ϕ 2 on Y is equivalently given by a combinatorial map Y → K(X, R) where K(X, R) is the standard 2-complex with respect to the group presentation G = X|R (with one vertex, an edge for each generator and its inverse, and a face for each relator and its inverse). We denote briefly Y = (V, E, F, ϕ 1 , ϕ 2 ).

A van Kampen diagram D is a finite, planar (embedded in the plan) and simply connected van Kampen 2-complex. Its boundary length |∂D| is the length of a boundary path, passing once by every edge adjacent to one face and twice by every edge adjacent to zero face.

A pair of faces in a van Kampen 2-complex is called reducible if they have the same relator label and there is a common edge on their boundaries at the same position. A van Kampen 2-complex is called reduced if there is no reducible pair of faces.

Reduction degree and Complexity

Let us define the reduction degree of a van Kampen 2-complex and the complexity of a 2-complex.

The reduction degree of a non-reduced van Kampen diagram Y = (V, E, F, ϕ 1 , ϕ 2 ) with respect to a group presentation X|R is the total number of geometric edges causing reducible pair of faces, counted with multiplicity: for any edge e ∈ E, any relator r ∈ R and any integer j, we count the number of faces f ∈ F labeled by r and having e as the j-th boundary edge. If this number is k, we add (k -1) + to the reduction degree where (•) + is the positive part function.

Definition 2.1 (Reduction degree, [GM18, Definition 2.5]). Let Y = (V, E, F, ϕ 1 , ϕ 2 ) be a van Kampen 2-complex of a group presentation G = X|R . Let ℓ be the maximal boundary length of faces of Y . The reduction degree of Y is

Red(Y ) = e∈E r∈R 1≤j≤ℓ {f ∈ F | ϕ 2 (f ) = r, e is the j-th edge of ∂f } -1 + . It is not hard to see that a van Kampen 2-complex Y is reduced if and only if Red(Y ) = 0.
A maximal arc of a 2-complex is a reduced combinatorial path passing only by vertices of degree 2 whose endpoints are not of degree 2. The complexity of a 2complex encodes the number of maximal arcs with the number of faces.

Definition 2.2 (Complexity of a 2-complex). Let Y be a 2-complex. Let K > 0. We say that Y is of complexity K if the following three conditions hold:

• |Y | ≤ K.
• The number of maximal arcs of Y is bounded by K.

• For any face f of Y , the boundary path ∂f is divided into at most K maximal arcs.

Note that if D is a planar and simply connected 2-complex with |D| ≤ K, then the complexity of D is 6K. In fact, as the rank of its underlying graph is K, the number of its maximal arcs is at most 3K, and every boundary path is divided into at most 6K maximal arcs (an arc may be used twice).

Lemma 2.3. Let K > 0. For ℓ large enough, the number of 2-complexes of complexity K with face boundary lengths at most ℓ is bounded by ℓ 3K .

Proof. As there are at most K maximal arcs, the number of vertices of valency greater than 3 is at most 2K/3 ≤ K. There are at most K 2 ways to draw an arc connecting two of these vertices, every arc is with length at most ℓ. The number of underlying graphs is then at most (K 2 ) K ℓ K .

To attach K faces on a graph, we choose K loops passing by at most K arcs, there are at most (K 2 ) K 2 choices. There are at most (2ℓ) K ways to choose a starting point and an orientation for every face. The number of such 2-complexes is hence bounded by

(K 2 ) K ℓ K × (K 2 ) K 2 × (2ℓ) K ,
which is smaller than ℓ 3K if ℓ is large enough.

Remark 2.4. While the number of 2-complexes with a bounded complexity grows polynomially with the maximal face boundary length ℓ, it is not the case for 2-complexes with a bounded number of faces. Hence the related argument in Proof of Theorem 1.4 does not work. Actually, there exists van Kampen 2-complexes that contradicts the inequality of Theorem 1.4. For example, in [START_REF] Calegary | Random groups contain surface subgroups[END_REF], D. Calegari and A. Walker proved that at any density d < 1/2, there exists a number K depending only on d such that a.a.s. there is a reduced van Kampen 2-complex Y homeomorphic to a surface of genus O(ℓ) in G ℓ (m, d) with at most K faces. Since every edge is adjacent to two faces in a surface, we have

|Y (1) | ≤ 1 2 |Y |ℓ, while we expect that |Y (1) | ≥ 1 -d -s 2 > 1 2 |Y |ℓ.

Abstract van Kampen 2-complexes

Let (G ℓ (m, d)) be a sequence of random groups at density d, defined by

G ℓ (m, d) = x 1 , . . . , x m |R ℓ .
Recall that B ℓ is the set of all cyclically reduced words of length at most ℓ and |B e ll| = (2m -1) ℓ+O(1) . Let

0 < ε < 1 -d. Since log |B ℓ | |R ℓ | converges
in probability to the constant d, the probability event

Q ℓ := (2m -1) (d-ε 4 ℓ) ≤ |R ℓ | ≤ (2m -1) (d+ ε 4 ℓ) is a.a.s. true (cf. [Tsa21] Proposition 1.8).
If we consider the Bernoulli density model that the events {r ∈ R ℓ } through r ∈ B ℓ are independent of the same probability (2m -1) (d-1)ℓ , it is obvious that we have Pr(r 1 , . . . , r k ∈ R ℓ ) = (2m -1) k(d-1)ℓ for distinct r 1 , . . . , r k in B ℓ . In the permutation invariant density model, we have the following corresponding proposition, which is a variant of [Tsa21] Lemma 3.10. Proposition 2.5. Let r 1 , . . . , r k be pairwise different relators in B ℓ . We have

Pr (r 1 , . . . , r k ∈ R ℓ | Q ℓ ) ≤ (2m -1) k(d-1+ ε 2 )ℓ .
Abstract van Kampen 2-complexes, as abstract van Kampen diagrams introduced by Ollivier in [START_REF] Ollivier | Sharp phase transition theorems for hyperbolicity of random groups[END_REF], is a structure between 2-complexes and van Kampen 2-complexes that helps us solve 2-complex problems in random groups. Recall that since isolated edges do not affect fillability, we will only consider finite 2-complexes without isolated edges.

Definition 2.6 (Abstract van Kampen 2-complex). An abstract van Kampen 2-complex Y is a 2-complex (V, E, F ) with a labeling function on faces by integer numbers and their inverses ϕ 2 :

F → {1, 1 -, 2, 2 -, . . . , k, k -} such that ϕ 2 (f -1 ) = ϕ 2 (f ) -. We denote simply Y = (V, E, F, ϕ 2 ).
By convention (i -) -= i. The integers {1, . . . , k} are called abstract relators. Similar to a van Kampen diagram, a pair of faces f, f ′ ∈ F is reducible if they are labeled by the same abstract relator, and they share an edge at the same position of their boundaries. An abstract diagram is called reduced if there is no reducible pair of faces. Let ℓ be the maximal boundary length of faces. The reduction degree of Y 2-complex can be similarly defined as

Red( Y ) = e∈E 1≤i≤k 1≤j≤ℓ {f ∈ F | ϕ 2 (f ) = i, e is the j-th edge of ∂f } -1 + .
We say that an abstract van Kampen 2-complex with k abstract relators Y = (V, E, F, ϕ 2 ) is fillable by a group presentation G = X|R (or by a set of relators R) if there exists k different relators r 1 , . . . , r k ∈ R such that the construction ϕ 2 (f ) := r ϕ2(f ) gives a van Kampen 2-complex Y = (V, E, F, ϕ 1 , ϕ 2 )3 of G. The k-tuple of relators (r 1 , . . . , r k ), or the van Kampen 2-complex Y , is called a filling of Y . As we picked different relators for different abstract relators, if Y is a filling of Y , then Red(Y ) = Red( Y ), and Y is reduced if and only if Y is reduced.

2 1 f illing ----→ Y r 2 r 1 Y Figure 2: Filling an abstract van Kampen 2-complex.
Denote ℓ i the length of the abstract relator i for 1 ≤ i ≤ k. Let ℓ = max{ℓ 1 , . . . , ℓ k } be the maximal boundary length of faces. The pairs of integers (i, 1), . . . , (i, ℓ i ) are called abstract letters of i. The set of abstract letters of Y is then a subset of the product set {1, . . . , k} × {1, . . . , ℓ}. The geometric edges of Y are decorated by abstract letters and directions: Let f ∈ F labeled by i and let e ∈ E at the j-th position of ∂f . The geometric edge e is decorated, on the side of f , by an arrow indicating the direction of e and the abstract letter (i, j). The number of decorations on a geometric edge is the number of its adjacent faces with multiplicity (an edge may be attached twice by the same face). Definition 2.7 (free-to-fill). An abstract letter (i, j) of D is free-to-fill if, for any edge e decorated by (i, j), it is the minimal decoration on e.

Denote α i the number of faces labeled by the abstract relator i and η i the number of free-to-fill edges of i. We have the following estimation.

Lemma 2.8. Let Y = (V, E, F, ϕ 2 ) be an abstract van Kampen 2-complex with k abstract relators. Then

k i=1 α i η i ≤ | Y (1) | + Red( Y ).
Proof. Denote E the set of geometric edges and F the set of geometric faces. For any geometric edge e, an adjacent face f from which the decoration is minimal is called a preferred face of e. For any face f , let E f be the set of geometric edges e on its boundary such that f is a preferred face of e. Note that an edge will never be counted twice as the decorations given by one face are all different. According to Definition 2.7, for any face f with ϕ 2 (f ) = i, we have

η i ≤ |E f |. Hence, k i=1 α i η i ≤ f ∈F |E f |.
Denote Red(e) the reduction degree caused by the edge e. That is,

Red(e) := 1≤i≤k 1≤j≤ℓ {f ∈ F | ϕ 2 (f ) = i, e or e -1 is the j-th edge of ∂f } -1 + ,
so that the number of preferred faces of e is bounded by 1 + Red(e). Hence,

f ∈F |E f | ≤ e∈E 1 + Red(e) = | Y (1) | + Red( Y ).
Probability of filling. We shall estimate the probability that an abstract van Kampen 2-complex Y is fillable by a random group G ℓ (m, d). This step is the key to prove Theorem 1.4. Recall that

Q ℓ := (2m -1) (d-ε 4 ℓ) ≤ |R ℓ | ≤ (2m -1) (d+ ε 4 ℓ
) is an a.a.s. true probability event.

Lemma 2.9. Let Y be an abstract van Kampen 2-complex with k abstract relators. We have

Pr Y is fillable by G ℓ (m, d) Q ℓ ≤ 2m 2m -1 k (2m -1) k i=1 (ηi+(d-1+ ε 2 )ℓ) .
Proof. Let us estimate the number of fillings of Y . For every free-to-fill abstract letter (i, j), there are at most 2m ways to fill a generator if j = 1, at most (2m -1) ways to fill if j = 1 for avoiding reducible word. As there are η i free-to-fill abstract letters on the i-th abstract relator, there are at most 2m(2m -1) ηi-1 ways to fill it. So there are at most k i=1 2m(2m -1) ηi-1 ways to fill Y . Let Y be a van Kampen 2-complex, which is a filling of Y . The 2-complex Y is labeled by k different relators in B ℓ , denoted r 1 , . . . , r k . By lemma 2.5,

Pr (Y is a 2-complex of G ℓ (m, d) | Q ℓ ) = Pr (r 1 , . . . , r k ∈ R ℓ | Q ℓ ) ≤ (2m -1) k(d-1+ ε 2 )ℓ .
Hence

Pr Y is fillable by G ℓ (m, d) Q ℓ ≤ Y fills Y Pr (Y is a 2-complex of G ℓ (m, d) | Q ℓ ) ≤ k i=1 2m(2m -1) ηi-1 (2m -1) k(d-1+ ε 2 )ℓ ≤ 2m 2m -1 k (2m -1) k i=1 (ηi+(d-1+ ε 2 )ℓ) .
Lemma 2.10. Let Y be an abstract van Kampen 2-complex with k abstract relators. Suppose that Y does not satisfy the inequality given in Theorem 1.4, i.e.

| Y (1) | + Red( Y ) < (1 -d -ε)| Y |ℓ, then Pr Y is fillable by G ℓ (m, d) Q ℓ ≤ 2m 2m -1 (2m -1) -ε 2 ℓ .
Proof. Let Y i be the sub-2-complex of Y consisting of faces labeled by the i first abstract relators. Denote

P i = Pr Y i is fillable by G ℓ (m, d) Q ℓ .
Apply lemma 2.9 on Y i , we have

P i ≤ 2m 2m -1 i (2m -1) i j=1 (ηj+(d-1+ ε 2 )ℓ) .
Note that if Y is fillable by G ℓ (m, d) then its sub 2-complex Y i is fillable by the same group. So for any

1 ≤ i ≤ k, log 2m-1 (P k ) ≤ log 2m-1 (P i ) ≤ i j=1 η j + d -1 + ε 2 ℓ + log 2m-1 2m 2m -1 .
Without loss of generality, suppose that

α 1 ≥ α 2 ≥ • • • ≥ α k . Note that log 2m-1 (P k ) is negative and α 1 ≤ | Y |, so | Y | log 2m-1 (P k ) ≤ α 1 log 2m-1 (P k ). By Abel's summa- tion formula, with convention α k+1 = 0, | Y | log 2m-1 (P k ) ≤ α 1 log 2m-1 (P k ) = k i=1 (α i -α i+1 ) log 2m-1 (P k ) ≤ k i=1 (α i -α i+1 ) i j=1 η i + d -1 + ε 2 ℓ + log 2m-1 2m 2m -1 = k i=1 α i η i + d -1 + ε 2 ℓ + log 2m-1 2m 2m -1 = k i=1 α i η i + k i=1 α i d -1 + ε 2 ℓ + log 2m-1 2m 2m -1 . Note that k i=1 α i = | Y |.
By Lemma 2.8 and the hypothesis of the current lemma,

k i=1 α i η i ≤ | Y (1) | + Red( Y ) < (1 -d -ε)| Y |ℓ.
Hence,

| Y | log 2m-1 (P k ) ≤ (1 -d -ε)| Y |ℓ + | Y | d -1 + ε 2 ℓ + log 2m-1 2m 2m -1 ≤ | Y | - ε 2 ℓ + log 2m-1 2m 2m -1 .

Proof of Theorem 1.4

Under the condition

Q ℓ := {(2m -1) (d-ε 4 ℓ) ≤ |R ℓ | ≤ (2m -1) (d+ ε 4 ℓ) }
, the probability that there exists a van Kampen 2-complex of complexity K of G ℓ (m, d) satisfying the inverse inequality

|Y (1) | + Red(Y ) < (1 -d -ε)|Y |ℓ ( * )
is bounded by

Y of complexity K, satisfying ( * ) Pr Y is fillable by G ℓ (m, d) Q ℓ .
By Lemma 2.3 and the face that there at most K 2K ways to label a 2-complex with K faces by abstract relators {1 ± , . . . , K ± }, there are at most ℓ 3K × K 2K terms in the sum. By Lemma 2.10, every term is bounded by 2m 2m-1 (2m -1) -ε 2 ℓ . So the sum is smaller than

ℓ 3K K 2K 2m 2m -1 (2m -1) -ε 2 ℓ ,
which converges to 0 as ℓ → ∞. By definition Pr(Q ℓ ) ---→ ℓ→∞ 1, so the probability that there exists a van Kampen 2-complex of G ℓ (m, d) of complexity K satisfying ( * ) converges to 0 as ℓ goes to infinity. That is to say, a.a.s. every van Kampen diagram of G ℓ (m, d) of complexity K satisfies the inequality

|Y (1) | + Red(Y ) ≥ (1 -d -ε)|Y |ℓ.
Closed surfaces. Recall that a 2-complex Y without isolated edges is called contractible if there is an edge of Y that is adjacent to one single face. If a 2-complex Y is not contractible, then each of its edge is adjacent to at least 2 faces, and we have |Y (1) | ≤ 1 2 |Y |ℓ where ℓ is the maximal boundary length of faces. As this contradicts the inequality of Theorem 1.4 for any density d < 1/2, we have the following proposition.

Proposition 2.11. Let (G ℓ (m, d)) be a sequence of random groups with m ≥ 2 generators at density d. For any d < 1/2 and K > 0, a.a.s. every reduced van Kampen 2-complex of complexity K of G ℓ (m, d) can be contracted to a graph.

Phase transition for the existence of van Kampen 2complexes

In this section, we work on the proof of Theorem 1.5.

Motivation and a counterexample. Let (G ℓ (m, d)) be a sequence of random groups at density d. We are interested in the converse of Theorem 1.4 without the reduction part: if a 2-complex Y ℓ with bounded complexity satisfies the inequality

|Y (1) ℓ | ≥ (1 -d + s)|Y ℓ |ℓ
with some s > 0, does there exist a face labeling by relators and an edge labeling by generators, so that Y ℓ becomes a reduced van Kampen 2-complex of G ℓ (m, d)?

The motivation for this question comes from the well-known phase transition at

density d = λ 2 , mentioned in [Gro93, p. 274]: if d < λ 2 then a.a.s. G ℓ (m, d) has the C ′ (λ) small cancellation condition; while if d > λ 2 then a.a.s. G ℓ (m, d) does not have C ′ (λ).
The first assertion is a simple application of Theorem 1.3. For the second assertion, we need to show that a.a.s. there exists a van Kampen 2-complex D of G ℓ (m, d) with exactly 2 faces of boundary length ℓ, sharing a common path of length at least λℓ (Figure 4).

The first detailed proof of such an existence is given in [BNW20, Theorem 2.1], using an analog of the probabilistic pigeonhole principle. Another proof is given in [START_REF] Tsai | Density of random subsets and applications to group theory[END_REF]Theorem 1.4]. An intuitive explanation using the "dimension reasoning" is given in [START_REF] Ollivier | A January 2005 invitation to random groups[END_REF] p.30: The dimension of the set of couples R ℓ × R ℓ is 2dℓ. Sharing a common subword of length L imposes L equations, so the "dimension" of the set of couples of relators sharing a common subword of length λℓ is 2dℓ -λℓ. then there will exist such a couple because the dimension will be positive. However, this argument is not true for any 2-complex in general. Here is a counterexample:

At density d = 0.4, let (D ℓ ) be a sequence of 2-complexes where D ℓ is given in Figure 5. The given inequality is satisfied because

|D (1) ℓ | = 1.9ℓ > 1.8ℓ = (1 - d)|D ℓ |ℓ. However, the sub-diagram D ′ ℓ gives |D ′(1) ℓ | = 1.1ℓ < 1.2ℓ = (1 -d)|D ′ ℓ |ℓ),

Geometric form and Critical density

Let us define the geometric form of 2-complexes and the critical density of a geometric form. To simplify the notations, for a 2-complex Y = (V, E, F ), we denote Edge(Y ) as the set of geometric edges of Y and e instead of e for geometric edges. A sequence of 2-complexes (Y ℓ ) is briefly said to be of the same geometric form if the geometric form (Y, λ) is not specified. Note that the boundary length of every face Hence, the condition " dens c (Y ) + d > 1" is equivalent to the following statement: Given s > 0, for ℓ large enough, every sub-2-complex Z ℓ of Y ℓ satisfies

f of Y ℓ is at most ℓ. If Z is a sub-2-complex of Y , we denote Z ≤ Y . By convention, if (Z ℓ )

|Z

(1)

ℓ | ≥ (1 -d + s)|Z ℓ |ℓ.
This argument shows that the second assertion of Theorem 1.5 is equivalent to Corollary 1.6.

Proof of Theorem 1.5 (i). We will use Theorem 1.4 without the reduction part. Let (G ℓ (m, d)) be a sequence of random groups with m generators at density d. Recall that a 2-complex Y ℓ is said to be fillable by G ℓ (m, d) if there exists a reduced van Kampen 2-complex of G ℓ (m, d) whose underlying 2-complex is Y ℓ .

Let (Y, λ) be a geometric form of 2-complexes with dens c Y + d < 1. Let (Y ℓ ) be a sequence of 2-complexes of the geometric form (Y, λ). We shall prove that a.a.s. the 2-complex Y ℓ is not fillable by the random group G ℓ (m, d). By the definition of critical density, there exists a sub-2-complex Z ≤ Y satisfying dens Z + d < 1. Let (Z ℓ ) be the sequence of 2-complexes of the geometric form (Z, λ |Z ). We shall prove that a.a.s. Z ℓ is not fillable by G ℓ (m, d).

Let ε > 0 such that dens Z = 1 -d -3ε. By definition,

lim ℓ→∞ |Z (1) ℓ | |Z ℓ |ℓ = 1 -d -3ε, so for ℓ large enough, |Z (1)
ℓ | ≤ (1 -d -2ε)|Z ℓ |ℓ < (1 -d -ε)|Z ℓ |ℓ.
The complexity of Z ℓ is K = max |Z|, |Z (1) |, max{ 1 λe | e ∈ Edge(Z)} , independent of ℓ. By Theorem 1.4 with ε and K given above, a.a.s. every van Kampen

2-complex Z ℓ of G ℓ (m, d) of complexity K should satisfy |Z (1) ℓ | ≥ (1 -d -ε)|Z ℓ |ℓ.
Hence, a.a.s. the given 2-complex Z ℓ is not fillable by G ℓ (m, d), which implies that a.a.s. Y ℓ is not fillable by G ℓ (m, d).

The multidimensional intersection formula for random subsets

To prove the second assertion of Theorem 1.5, we need the multidimensional intersection formula for random subsets with density, introduced in [Tsa21, Section 3].

Recall that B ℓ is the set of cyclically reduced words of X ± m = {x ± 1 , . . . , x ± m } of length at most ℓ, and that |B ℓ | = (2m -1) ℓ+o(ℓ) . Let k ≥ 1 be an integer. Denote B (k) ℓ as the set of k-tuples of pairwise distinct relators (r 1 , . . . , r k ) in B ℓ . Such a notation can be used for any set or any random set.

Note that [START_REF] Gromov | Asymptotic invariants of infinite groups[END_REF]p.272] and [Tsa21, Definition 1.5]). That is to say, |Y ℓ | = (2m -1) αkℓ+o(ℓ) . Definition 3.4 (Self-intersection partition, [Tsa21, Definition 3.4]). Let (Y ℓ ) be a sequence of fixed subsets of the sequence (B

|B (k) ℓ | = (2m -1) kℓ+o(ℓ) . Recall that a sequence of fixed subsets (Y ℓ ) of the sequence (B (k) ℓ ) is called densable with density α ∈ {-∞} ∪ [0, 1] if the se- quence of real numbers (log |B (k) ℓ | |Y ℓ |) converges to α (see
(k) ℓ ). Let 0 ≤ i ≤ k be an integer. The i-th self-intersection of Y ℓ is S i,ℓ := {(x, y) ∈ Y 2 ℓ | |x ∩ y| = i}
where |x ∩ y| is the number of common elements between the sets x = (r 1 , . . . , r k ) and y = (r ′ 1 , . . . , r ′ k ). with density α. Let S i,ℓ with 0 ≤ i ≤ k be its self-intersection partition. Let d > 1 -α. We say that (Y ℓ ) satisfies the d-small self-intersection condition if, for every

The family of subsets {S

i,ℓ | 0 ≤ i ≤ k} is a partition of Y 2 ℓ , called the self- intersection partition of Y ℓ . Note that (S i,ℓ ) ℓ∈N is a sequence of subsets of the sequence (B (k) ℓ ) 2 ℓ∈N , with density smaller than dens (B (k) ℓ ) 2 Y 2 ℓ = dens (B (k) ℓ ) (Y ℓ ).
1 ≤ i ≤ k -1, dens (B (k) ℓ ) 2 (S i,ℓ ) < α -(1 -d) × i 2k .
Theorem 3.6 (Multidimensional intersection formula, [Tsa21, Theorem 3.6]). Let (R ℓ ) be a sequence of permutation invariant random subsets of (B ℓ ) of density d. Let

(Y ℓ ) be a sequence of fixed subsets of B (k) ℓ of density α > 1 -d. If (Y ℓ ) satisfies the d-small self intersection condition, then the sequence of random subsets (Y ℓ ∩ R (k) ℓ ) is densable with density α + d -1.
In particular, a.a.s. the random subset

Y ℓ ∩ R (k) ℓ of B (k) ℓ
is not empty.

Proof of Theorem 1.5 (ii)

Let (Y ℓ ) be a sequence of 2-complexes of the same geometric form (Y, λ) with k faces.

In the following, we denote Y ℓ as the set of pairwise distinct relators in B ℓ that fills Y ℓ , which is a subset of

B (k) ℓ . Let (G ℓ (m, d)) be a sequence of random groups at density d, defined by G ℓ (m, d) = X m |R ℓ where (R ℓ ) is a sequence of random subsets with density d. The intersection Y ℓ ∩ R (k) ℓ is hence the set of k-tuples of pairwise distinct relators in R ℓ that fills Y ℓ .
We want to prove that this intersection is not empty, so that Y ℓ is fillable by G ℓ (m, d). According to Theorem 3.6, it remains to prove that if dens c Y > 1 -d, then the sequence (Y ℓ ) is densable and satisfies the d-small self intersection condition.

We will prove in Lemma 3.8 that (Y ℓ ) is densable with density exactly dens(Y ), and in Lemma 3.9 that it satisfies the d-small self intersection condition.

Lemma 3.7. Let Y ℓ be the set of k-tuples of relators in B ℓ that fills Y ℓ , not necessarily pairwise distinct. If Y ℓ is fillable by B ℓ , then

dens (B k ℓ ) (Y ℓ ) = dens Y.
Proof. We shall estimate the number |Y ℓ | by counting the number of labelings on edges of Y ℓ that produce van Kampen 2-complexes with respect to all possible relators B ℓ . We start by filling edges in the neighborhoods of vertices that are originally vertices of the geometric form Y (before dividing). Consider the set of oriented edges of Y ℓ starting at some vertex that is originally a vertex of Y before dividing. A vertex labeling is a labeling on these edges by X ± m that does not produce any reducible pair of edges on face boundaries: for every pair of different edges e 1 , e 2 starting at the same vertex, if they are labeled by the same generator x ∈ X ± m , then the path e -1 1 e 2 is not cyclically part of any face boundary loop. Since the 2-complex Y ℓ is fillable, the set of vertex labelings is not empty. Denote C ≥ 1 as the number of vertex labelings of Y ℓ .

As m ≥ 2 and ⌊λ e ℓ⌋ ≥ 3 for ℓ large enough, if there exists a vertex labeling, then the other edges of Y ℓ can be completed as a van Kampen 2-complex of B ℓ , and the number C depends only on the geometric form Y .

To label the remaining ⌊λℓ⌋ -2 edges on the arc divided from the edge e ∈ Edge(Y ), there are 2m -1 choices for the first ⌊λℓ⌋ -3 edges, and 2m -2 or 2m -1 choices for the last edge. So ℓ) . We have

C e∈Edge(Y ) (2m -1) ⌊λeℓ⌋-3 (2m -2) ≤ |Y ℓ | ≤ C e∈Edge(Y ) (2m -1) ⌊λeℓ⌋-2 . Recall that k = |Y | = |Y ℓ | and that |B k ℓ | = (2m -1) kℓ+o(
dens (B k ℓ ) (Y ℓ ) = e∈Edge(Y ) λ e |Y | = dens Y. Lemma 3.8. If dens c Y > 1/2 and Y ℓ is fillable by B ℓ , then (Y ℓ ) is densable in (B (k) ℓ ) and dens (B (k) ℓ ) (Y ℓ ) = dens Y. Proof. Suppose that |Y | ≥ 2. The case |Y | = 1 is trivial. Let Z be a sub-2-complex of Y with exactly two faces f 1 , f 2 . As dens Z ≥ dens c Y > 1 2
, by Definition 3.2, we have

e∈Edge(Z) λ e > 1 2 |Z| = 1 ≥ |∂f 1 |.
Let Y Z ℓ be the set of fillings of Y ℓ by B ℓ such that the two faces of Z are filled by the same relator. By the same arguments of the previous lemma,

|Y Z ℓ | ≤ C(2m -1) |∂f1| e∈Edge(Y )\ Edge(Z) (2m -1) ⌊λeℓ⌋-2 , so dens (B k ℓ ) (Y Z ℓ ) ≤ 1 |Y |   e∈Edge(Y ) λ e +   |∂f 1 | - e∈Edge(Z) λ e     < e∈Edge(Y ) λ e |Y | = dens Y = dens (B k ℓ ) (Y ℓ ). Knowing that Y ℓ = Y ℓ Z<Y,|Z|=2 Y Z ℓ ,
we have

|Y ℓ | - Z<Y,|Z|=2 |Y Z ℓ | ≤ |Y ℓ | ≤ |Y ℓ |.
There are |Y | 2 terms in the sum, in every term we have dens

(B k ℓ ) (Y Z ℓ ) < dens (B k ℓ ) (Y ℓ )
, so (cf. [Tsa21, Proposition 2.7 and Proposition 2.8])

dens (B k ℓ ) (Y ℓ ) = dens (B k ℓ ) (Y ℓ ).
Together with Lemma 3.7, we have dens

(B k ℓ ) (Y ℓ ) = dens Y . As dens (B k ℓ ) (B (k) ℓ ) = 1, we get dens (B (k) ℓ ) (Y ℓ ) = dens Y.
Lemma 3.9. Suppose that dens c Y > 1 -d. Let S i,ℓ be the i-th self intersection of the set Y ℓ . We have

dens (B (k) ℓ ) 2 (S i,ℓ ) < dens Y -(1 -d) × i 2k . Proof. Let Z, W be two sub-2-complexes of Y with |Z| = |W | = i < k = |Y |. Let (Z ℓ ), (W ℓ )
be the corresponding sequences of 2-complexes of the geometric forms Z and W respectively. Denote S ℓ (Z, W ) as the set of pairs of pairwise distinct fillings ((r 1 , . . . , r k ), (r ′ 1 . . . , r ′ k )) of Y ℓ by all possible relators B ℓ such that, the i relators in the first filling (r 1 , . . . , r k ) corresponding to Z ℓ are identical to the i relators in the second filling (r ′ 1 , . . . , r ′ k ) corresponding to W ℓ , and that the remaining 2k -2i relators are pairwise different, not repeating the relators in Z ℓ and W ℓ .

Let us estimate the cardinality |S ℓ (Z, W )|. Firstly, fill the k-tuple (r 1 , . . . , r k ) so the i relators in the next k-tuple (r ′ 1 , . . . , r ′ k ) corresponding to the sub 2-complex W ℓ is determined. There are at most i! choices for ordering these i relators. To fill the remaining k -i relators in (r ′ 1 , . . . , r ′ k ), by the same arguments of Lemma 3.7, we get

|S ℓ (Z, W )| ≤ |Y ℓ | × i! × C e∈Edge(Y )\ Edge(W ) (2m -1) ⌊λeℓ⌋-2 .
Recall that the density of Y is defined by

dens (B (k) ℓ ) 2 (S ℓ (Z, W )) ≤ 1 2k   e∈Edge(Y ) λ e + e∈Edge(Y )\ Edge(W ) λ e   = 1 2k   2 e∈Edge(Y ) λ e - e∈Edge(W ) λ e   = dens Y - i 2k dens W < dens Y - i 2k (1 -d). Note that S i,ℓ = Z<Y,W <Y,|Z|=|W |=i S ℓ (Z, W ).
It is a union of k i 2 subsets of densities strictly smaller than dens Y -i 2k (1 -d). According to [Tsa21, Proposition 2.7], we have

dens (B (k) ℓ ) 2 (S i,ℓ ) < dens Y - i 2k (1 -d).
This completes the proof of Theorem 1.5.

Phase transitions for small cancellation conditions

Let us recall small cancellation notions in [LS77, p.240]. A piece with respect to a set of relators is a cyclic sub-word that appears at least twice. A group presentation satisfies the C ′ (λ) small cancellation condition for some 0 < λ < 1 if the length of a piece is at most λ times the length of any relator that it appears. It satisfies the C(p) small cancellation condition for some integer p ≥ 2 if no relator is a product of fewer than p pieces.

The C ′ (λ) condition. Let (G ℓ (m, d)) be a sequence of random groups at density d. It is known that there is a phase transition at density d = λ/2 for the C ′ (λ) condition (see [START_REF] Gromov | Asymptotic invariants of infinite groups[END_REF]p.274], [BNW20, Theorem 2.1] and [Tsa21, Theorem 1.4]). We give here a much simpler proof using Theorem 1.5. The C(p) condition. We shall prove by Theorem 1.5 that for random groups with density, there is a phase transition at density 1/(p + 1) for the C(p) condition. (ii) Consider a geometric form with p + 1 faces, one of the faces is placed in the center, having p edges of length 1/p, such that every edge is attached by another face with two edges of lengths 1/p and 1 -1/p. There are no other attachments (Figure 9).

The density of Y is 1+p(1-1/p)

  Definition 1.1 ([Gro93, p.272], [Tsa21, Definition 1.5 and Definition 2.5]). A sequence of random subsets (R ℓ ) of the sequence of sets (B ℓ ) is called densable with density d ∈ {-∞} ∪ [0, 1] if the sequence of random variables dens B ℓ (R ℓ ) := log |B ℓ | (|R ℓ |) converges in probability to the constant d.

  Definition 1.2 ([Gro93, p.273], [Tsa21, Definition 4.1]). A sequence of random groups (G ℓ (m, d)) with m generators at density d is defined by G ℓ (m, d) = X m |R ℓ

  Theorem 1.3 ([Gro93, p.274], [Oll04, Chapter 2]). Let (G ℓ (m, d)) be a sequence of random groups with m ≥ 2 generators at density d. For any ε > 0 and K > 0, a.a.s. every reduced van Kampen diagram D of G ℓ (m, d) with |D| ≤ K satisfies the isoperimetric inequality |∂D| ≥ (1 -2d -ε)|D|ℓ.

Figure 1 :

 1 Figure 1: A reducible pair of faces.

Figure 3 :

 3 Figure 3: A geometric edge decorated by two abstract letters.

  Figure 4: A van Kampen diagram denying the C ′ (λ) condition.

  which contradicts the isoperimetric inequality of Theorem 1.4 and can not be a van Kampen diagram of G ℓ (m, d).

Figure 5 :

 5 Figure 5: A 2-complex that satisfies the isoperimetric inequality with a sub-2-complex that does not.

  Definition 3.1. A geometric form of 2-complexes is a couple (Y, λ) where Y = (V, E, F ) is a finite connected 2-complex without isolated edges, and λ is a length labeled on edges defined by λ : Edge(Y ) →]0, 1], e → λ e , such that for every face f of Y , the boundary length |∂f | is bounded by 1.A sequence of 2-complexes (Y ℓ ) is called of the geometric form (Y, λ) if Y ℓ is obtained from Y by dividing every edge e of Y into ⌊λ e ℓ⌋ 4 edges of length 1.

  is a sequence of 2-complexes of the geometric form (Z, λ |Z ), we have Z ℓ ≤ Y ℓ for any integer ℓ. Definition 3.2. Let (Y, λ) be a geometric form of 2-complexes. The density of Y is dens(Y ) := e∈Edge(Y ) λ e |Y | . The critical density of Y is dens c (Y ) := min Z≤Y {dens(Z)}. The intuition of this definition can be found in Lemma 3.8: the density of Y is actually the density of all possible van Kampen 2-complexes that fill Y ℓ . Remark 3.3. Taking Definition 3.2 and Definition 3.1 together, we have dens(Y ) = e∈Edge(Y ) λ e |Y | = lim ℓ→∞ e∈Edge(Y ) ⌊λ e ℓ⌋ |Y ℓ |ℓ = lim ℓ→∞ |Y (1) ℓ | |Y ℓ |ℓ .

Definition 3. 5

 5 (d-small self-intersection condition, [Tsa21, Definition 3.5]). Let (Y ℓ ) be a sequence of fixed subsets of B (k) ℓ

  ) λ e , and that dens W ≥ dens c Y > 1 -d by Definition 3.2. Together with the hypothesis dens c Y > 1 -d, we have

Proposition 4. 1 .

 1 Let 0 < λ < 1. Let (G ℓ (m, d)) be a sequence of random groups at density d. There is a phase transition at density d = λ/2: (i) If d < λ/2, then a.a.s. G ℓ (m, d) satisfies C ′ (λ).

(r 2 r 1 Figure 6 :Figure 7 :

 167 Figure 6: A van Kampen 2-complex constructed from a C ′ (λ) group.

Proposition 4. 2 .

 2 Let p ≥ 2 be an integer. Let (G ℓ (m, d)) be a sequence of random groups at density d. There is a phase transition at density d = 1/(p + 1): (i) If d < 1/(p + 1), then a.a.s. G ℓ (m, d) satisfies C(p).

(

  ii) If d > 1/(p + 1), then a.a.s. G ℓ (m, d) does not satisfy C(p). Proof. (i) Let us prove by contradiction. Suppose that a.a.s. G ℓ (m, d) does not satisfy C(p). That is to say, a.a.s. there exists a reduced van Kampen diagram D with (p + 1) faces, one face is placed in the center, attached by the other p faces on the whole boundary, and there is no other attachments (Figure 8).

Figure 8 :

 8 Figure 8: A van Kampen 2-complex constructed from a C(p) group. We have |D| = p + 1 and |D (1) | ≤ pℓ. Let ε = 1 p+1 -d /2, we have |D (1) | ≥ (1 -d -ε)|D|ℓ, which contradicts Theorem 1.4 .

p+1=Figure 9 :

 9 Figure 9: The geometric form for the C(p) condition.

(

  ii) If d > 1/(2p + 2), then a.a.s. G ℓ (m, d) does not satisfy B(2p).

Figure 10 :

 10 Figure 10: The geometric form for the B(2p) condition.

Note that every van Kampen diagram composed of relators of lengths at most ℓ satisfies

2|D (1) | -|∂D| ≤ |D|ℓ, so the given inequality implies the isoperimetric inequality.

A model that the relator length ℓ =

is fixed, and we are interested in asymptotic behaviors when the number of generators m goes to infinity.

Note that the edge labeling ϕ 1 is determined by the face labeling ϕ 2 as there is no isolated edges.

We can replace ⌊λeℓ⌋ by any function with λℓ + o(ℓ) and slightly smaller than λℓ. Note that the sum of edge lengths on every face boundary of Y ℓ is at most ℓ