Sébastien Fregonese

Chhandak Mukherjee

Holger Rücker

Pascal Chevalier

Gerhard Fischer

Didier Céli

Marina Deng

Marine Couret

François Marc

Cristell Maneux

Thomas Zimmer

Electro-thermal limitations and device degradation of SiGe HBTs with emphasis on circuit performance

Keywords: SiGe HBTs, avalanche breakdown, self-heating, limitations, reliability, safe operating area (SOA), HICUM compact model, Aging Laws I

This paper examines the performance of SiGe HBTs under DC and AC pulsed operating conditions beyond the breakdown voltage. Two state-of-the-art technologies are investigated. The limitations when biasing the transistor at or beyond peak fT are thermal only, thus penalizing deep trench architectures. Furthermore, the reliability implications are explored and their impact on device degradation is assessed.

INTRODUCTION

Today, SiGe BiCMOS is the technology of choice for demanding (sub) mm-wave applications like high-data rate wireless communications, optoelectronic data links, nextgeneration automotive radar sensors and future novel THz applications. Compared to III-V devices, SiGe HBTs possess attractive thermal and reliability properties as well as a much higher integration density coupled with an efficient BEOL architecture allowing the realization of transmission lines in micro-strip configuration as well as manifold additional passive devices. The only drawback is, however, the lower breakdown voltage (BV) of SiGe HBTs due to their smaller bandgap and the higher current gain  compared to most of the III-V HBTs. Thus, obtaining a high output power density of SiGe HBTbased power amplifier (PA) is a challenging design task.

To achieve this, several strategies are possible. 1-The maximum current swing can be increased by the use of PA-cells composed of multiple emitter finger configurations. The drawback here is related to the design of the input and output matching circuits due to the lower input and output impedances [START_REF] Grens | Reliability of SiGe HBTs for Power Amplifiers--Part I: Large-Signal RF Performance and Operating Limits[END_REF] of large area devices. 2-The maximum voltage swing can be increased by operating the device at higher collector voltage. Typically, DC collectorto-emitter breakdown voltage with base open (BVCEO) or DC collector-to-base breakdown voltage with open emitter (BVCBO) are used as the metrics for voltage limit of PA devices. In practical PA circuits, the RF envelope voltage can swing well beyond BVCEO without causing a failure. An analysis of output power swing limitations and DC breakdown is presented with a focus on biasing and temperature in [START_REF] Jain | DC and RF breakdown voltage characteristics of SiGe HBTs for WiFi PA applications[END_REF].

For both strategies, reliability is a concern.

Other strategies exist that are based on circuit design, which will not be detailed in this paper, but are mentioned for the sake of completeness: In order to increase Pout for high-speed HBTs with a relatively low BVCEO, VCE can be improved using a cascode circuit configuration. A cascode PA consists of a stack of HBTs in CB configuration on top of a CE configuration which can thus reduce the output voltage clipping. Note that the number of stacked transistors is limited by the substrate breakdown voltage. Connecting cascodes in parallel further increases the output power and reduces the collector current clipping of the PA [START_REF] Sakalas | Distortion analysis of CE and CB SiGe HBT power cells with fmax beyond 220 GHz for millimeterwave applications[END_REF]. Use of a cascode topology with a low terminating base impedance and minimum voltage-current waveform-overlap extends the VCE swing on the upper SiGe HBT in the cascode beyond its BVCBO, boosting output power and power added efficiency (PAE). [START_REF] Ju | An X-band inverse class-F SiGe HBT cascode power amplifier With harmonic-tuned output transformer[END_REF] Another option for mm-wave power amplifier design is the use of the Class-E architecture which is an attractive proposition due to its high theoretical collector efficiency as well as higher allowable voltage swing owing to non-overlapping voltage and current waveforms. Specifically, in HBTs the voltage swing can be as high as BVCBO as opposed to BVCEO that sets the limit in linear amplifiers [START_REF] Datta | Performance Limits, Design and Implementation of mm-Wave SiGe HBT Class-E and Stacked Class-E Power Amplifiers[END_REF]. Finally, pulsed systems such as short-time (ultra-) wideband power amplifiers (UWB-PA) provide essential advantages over continuous wave (CW) PAs in terms of size, weight, and power dissipation for a given output power requirement [START_REF] Rodenbeck | 50-W LTCC transmitter utilizing 28-V GaAs with integrated high-speed pulse modulation[END_REF]. These systems can be found in the fields of high-resolution imaging and sensing, precision position location, and wireless networking [START_REF] Rodenbeck | Techniques for the Analysis and Elimination of Transient Oscillations in Wideband and Ultra-Wideband Pulsed Power Amplifiers[END_REF].

This paper is divided into two parts: First, we will explore the maximum collector-to-emitter breakdown voltage of HBTs under test. The device operates at its best RF-performance (VBE at peak-fT). Pulsed DC and AC measurements are carried out. Two different state-of-the-art architectures are investigated and their sensitivity to self-heating is explored. In the second part, the device degradation due to operation close to the safeoperating-area (SOA) is analyzed and modelled. After a short presentation of the implemented aging compact HICUM model, the impact of the degradation on circuit building blocks are illustrated.

II. PULSED MODE OPERATION AT BREAKDOWN VOLTAGE AND BEYOND

The DC output characteristics of a bipolar transistor is represented by its IC (VCE, for different constant VBE values). When investigating these curves for high VCEs, a 1 st snapback (or flyback) behavior, i.e., the onset of a negative-differentialresistance (NDR) branch, is observed at medium VBE; it is worth noting that the NDR branch is measured by keeping IC constant while sweeping VBE and measuring VCE [START_REF] Spina | Influence of Concurrent Electrothermal and Avalanche Effects on the Safe Operating Area of Multifinger Bipolar Transistors[END_REF]. The 1 st snapback is one of the critical borders (edges?) for the safe operating area (SOA), since under VCE-controlled conditions the IC would sharply and significantly increase beyond that point; at high currents a 2 nd snapback occurs, after which a positive-differential-resistance (PDR) branch is restored. The well-known mechanisms described before is valid for medium VBE values only. For high VBEs, the measured IC-VCE curves show fully PDR behavior (i.e. the 1 st snapback does not occur) and are purely limited by self-heating. The SOA is limited by BVCBO at low VBE, and by the pinch-in effect at low-to-medium VBEs [START_REF] Jaoul | Measurement based accurate definition of the SOA edges for SiGe HBTs[END_REF].

On the other hand, the VBE region of interest is located at high VBEs, where the device operates close to the peak transit frequency (fT) or fMAX. In order to analyze the impact of selfheating, pulsed measurements are an ideal tool for in-depth investigation. In addition, it provides crucial information for the UWB-PA design. In the present study, pulse measurements are carried out with the pulsed DC analyzer Keithley 4200-SCS in combination with the VNA Rohde & Schwarz ZVA67 (with ZVAX extension unit). The PMU allows to create pulses larger than 60 ns with minimum rise and fall times of 20 ns [START_REF] Weisz | 80 ns/45 GHz Pulsed measurement system for DC and RF characterization of high speed microwave devices[END_REF]. Measurements with sufficient accuracy were obtained using a time window between 60% and 90% of the DC pulse duration.

These measurements have been performed on two state-ofthe-art technologies as presented in Fig. 1: Fig 1a shows a crosssectional view of the HBT used here that were from a development lot of a new generation of the 130 nm BiCMOS technology, SG13G3 [START_REF] Rücker | Device Architectures for High-speed SiGe HBTs[END_REF]. It shows an elevated extrinsic base (EEB) with a self-aligned emitter-base architecture created using the NSEG of the base. Its key features include: (i) partially mono-crystalline EEB regions are formed self-aligned to the emitter, facilitating low extrinsic base resistances RBX; (ii) the formation of the high-speed HBT device within a single active area without shallow trench isolation (STI) between the emitter and the collector contact regions, resulting in low RC and small collector-substrate junction areas. The HBTs feature peak fT/fMAX values of 470/650 GHz. The cross sectional view from Fig. 1b transistor shows the B55 technology from STMicroelectronics (ST). It presents the DPSA-SEG architecture, which is well-known for its very popular and fully self-aligned structure. The architecture exhibited state-of-theart performances of 320 GHz fT and 370 GHz fMAX for the 55nm SiGe BiCMOS [START_REF] Chevalier | A 55 nm triple gate oxide 9 metal layers SiGe BiCMOS technology featuring 320 GHz fT / 370 GHz fMAX HBT and High-Q Millimeter-Wave Passives[END_REF]. Although, it is not recommended for circuit applications due to transistor instability and reduced frequency performances, we have operated the transistors well beyond their SOA to gain a deeper understanding of the ultimate limit of these technologies and verify the accuracy of the electro-thermal model.

Results of DC pulsed measurements from both technologies are shown in Fig. 2. We can observe that (i) under pulsed conditions, the transistor can operate well beyond the SOA (maximum allowed VCE is 1.6 V for case (a) [START_REF] Rücker | Device Architectures for High-speed SiGe HBTs[END_REF] and 1.5 V for case (b) [START_REF] Chevalier | A 55 nm triple gate oxide 9 metal layers SiGe BiCMOS technology featuring 320 GHz fT / 370 GHz fMAX HBT and High-Q Millimeter-Wave Passives[END_REF]), (ii) a high impact of the pulse width on the collector current, which is directly linked to the dynamic self-heating, (iii) this impact is much more pronounced for the transistor presented in Fig. 2b compared to Fig. 2a.

Two main structural differences between both technologies, apart from the EEB in the case of the IHP technology, can be noted: (i) the fragmented emitter for IHP HBTs with 8 unit cells of 0.11 × 1.0 μm 2 , (ii) the presence of the deep trench isolation schemes in the ST technology. Deep trenches are known to confine the heat flux resulting in a higher thermal resistance. Additionally, the dynamic behavior is altered as can be observed from Fig. 2 where, for a given VBE, different pulse widths have been applied. The question now arises on how the transistor behaves for pulsed AC operation.

In Fig. 3, we show AC pulsed measured data, for a DC pulse width of 150ns and an RF pulse width of 50ns. The applied VBE voltage is 0.9V while the VCE varies between 1.0V, 1.5V and 2.0V. The results shown in Fig. 3 are quite unexpected: On the LHS of Fig. 3 (IHP), the small-signal parameters are much more sensitive compared to the figures of the RHS (ST). On the other hand, the pulsed DC voltages have a much higher impact on the collector current for the ST technology (RHS). These two results seem to contradict each other and need more explanation. When we examine the collector current density at VBE=0.9V, we recognize that for the IHP technology (LHS), JC is higher (up to two times for a VCE=2V). That implies, that even with a lower thermal resistance, due to the absence of deep trenches, the final device temperature is much higher making the small signal parameters more sensitive to a variation of the operating point (different VCEs), even under pulsed conditions. Here the minimum pulse widths is 150 ns and even for this short pulse width, self-heating is already noticeable (see also the pulsed DC curves at VBE=0.9V in Fig. 2). In fact, the absence of DTI leads to a very small self-heating (thermal?) time constant. Hence, despite the short pulses, the pulsed measurements are not adequate to reduce self-heating in the case of the IHP technology. In the case of ST, the pulsed measurements can indeed reduce the self-heating, especially the part due to DTI, which has a much higher time constant compared to the intrinsic part. Using the HICUM L2 compact model, we estimate that the reduction of self-heating due to pulsed measurement is up to 50 % for the case of the ST devices. This analysis is confirmed by HICUM L2 compact model simulation. In order to achieve a quite good agreement with simulation, as shown in Fig. 3, the temperature dependence of RTH requires particular attention for modeling, which underlines the dominance of the thermal effect. In Fig. 4, we present again the S11 and the small-signal current gain h21, this time only for one operating point (beyond BVCE0: VBE=0.9V and VCE=2V), but for different pulse widths.

Here we observe that the variation in the small-signal parameters is much more pronounced in the case of the ST technology (Fig 4 . b andd) compared to the IHP one (Fig. 4 a andc). We attribute this behavior again to the presence/absence of deep trenches. Without deep trenches (IHP) the heating is quite fast and the final steady thermal state is reached in a short time; whereas in the presences of deep trenches, a second/third time constant delays the heating mechanism. When comparing the results to simulation, we can observe that in both cases the results obtained using longer pulses tend to match with the DC-RF continuous wave simulation results. In this section, we have demonstrated the benefit of operating the device under pulsed conditions that allows the devices to enter the Safe Operating Area and thus to deliver high output power density. The downside of this approach is that reliability issues that are related to temperature and avalanche mechanisms [START_REF] Raghunathan | Bias-and Temperature-Dependent Accumulated Stress Modeling of Mixed-Mode Damage in SiGe HBTs[END_REF] may occur. We will investigate these issues in the following part of the paper.

III.

RELIABILITY ANALYSIS: FROM DEVICE TO CIRCUIT High base-collector voltage results in creation of electronhole pairs (avalanche mechanism) where the holes may give rise to hot-carrier degradation (HCD). The underlying mechanism of HCD is trap generation at the emitter-base (EB) spacer oxide interface through Si-H bond-breaking followed by non-ideal base current degradation [START_REF] Mukherjee | Hot Carrier Degradation in SiGeHBTs: A Physical and Versatile Aging Compact Model[END_REF]. The schematic in Fig. 5 (a) illustrates the HBT structure in which HC damage is located near the E-B spacer which is followed by subsequent Si-H bond breakage at the [START_REF] Mukherjee | Hot Carrier Degradation in SiGeHBTs: A Physical and Versatile Aging Compact Model[END_REF][START_REF] Cressler | Emerging SiGe HBT reliability issues for mixed-signal circuit applications[END_REF][START_REF] Kamrani | Microscopic Hot-Carrier Degradation Modeling of SiGe HBTs under Stress Conditions Close to the SOA limit[END_REF][START_REF] Jaoul | Analysis of a failure mechanism occurring in SiGe HBTs under mixed-mode stress conditions[END_REF] oxide interface and Hdiffusion into the oxide (Fig. 5(b)).

A.

Analytical model formulation The rate of bond dissociation is governed by a chemical interaction between the carriers and the passivated Si-H bond through generation and annihilation of traps. The interface-trap density, NT (t), increases with the net rate of reaction and the bond dissociation rate can be calculated following reactiondiffusion theory [START_REF] Jeppson | Negative bias stress of MOS devices at high electric fields and degradation of MNOS devices[END_REF] as follows, 𝑑𝑁 𝑑𝑡 = 𝐾 𝑁 -𝑁 (𝑡) -𝐾 𝑁 (𝑡)𝑁 (0, 𝑡)

KF is the rate constant of the forward reaction, i.e., generation of traps, KR is the rate constant of trap annihilation by hydrogen atoms, 𝑁 (𝑥, 𝑡) is the volumetric density of hydrogen at distance x of the Si/SiO2 interface and NF is the total number of available bonds that can break. From a compact model point of view [START_REF] Schröter | Compact hierarchical modeling of bipolar transistors with HICUM[END_REF], the trap creation at the Si/SiO2 interface (Fig. 5 (a)) can be represented by the change in a specific base current parameter, the peripheral B-E recombination saturation current, IREpS [START_REF] Couret | Scaled formulation for trap generation near the EB spacer oxide interface in SiGe HBTs[END_REF][START_REF] Mukherjee | A physical and versatile aging compact model for hot carrier degradationin SiGe HBTs under dynamic operating conditions[END_REF]. As it has been made evident in earlier works [START_REF] Mukherjee | Hot Carrier Degradation in SiGeHBTs: A Physical and Versatile Aging Compact Model[END_REF][START_REF] Couret | Scaled formulation for trap generation near the EB spacer oxide interface in SiGe HBTs[END_REF][START_REF] Mukherjee | A physical and versatile aging compact model for hot carrier degradationin SiGe HBTs under dynamic operating conditions[END_REF] that the region around the emitter periphery is solely impacted by HCD, the effect of degradation on the internal base-emitter current parameters can be neglected. In [START_REF] Couret | Scaled formulation for trap generation near the EB spacer oxide interface in SiGe HBTs[END_REF], we used TCAD simulations along with HICUM compact model simulations to correlate the increase of IREpS with that of the trap density at the spacer interface (Fig. 6(a)) for a given emitter perimeter, PE, through the following equation, 𝐼 (𝑡) = 𝛼𝑃 𝑁 (𝑡)

(2) In addition to this setup, HICUM AL V3.01 employs an additional fictitious transistor node for the oxide trap generation through the total oxide trap density 𝑁 (𝑡) (Fig. 6 (b)) which is governed by an exponential stress time dependence similar to the implementation of bulk trap in [START_REF] Mukherjee | A unified aging compact model for hot carrier degradation under mixed-mode and reverse E-B stress in complementary SiGe HBTs[END_REF]. Finally, in addition to trap generation and creation of H-atoms (Fig. 5 (a) and(b)) through Si-H bond dissociation, the H-diffusion is modeled by an N-stage RC ladder network terminated by a conductance G (Fig. 6 (c)) that represents the long-term diffusion scenario [START_REF] Couret | Scaled formulation for trap generation near the EB spacer oxide interface in SiGe HBTs[END_REF][START_REF] Mukherjee | A physical and versatile aging compact model for hot carrier degradationin SiGe HBTs under dynamic operating conditions[END_REF]. This R-C network considers all possible diffusion scenarios based on Fick's law of H-diffusion [START_REF] Fick | V. on liquid diffusion[END_REF].

B. Device Aging Simulation

Following an initial HICUM model calibration against experimental data using foundry-generated model cards, HICUM AL V3.01 simulations were validated against agingtests performed on SiGe NPN HBTs from Infineon Technologies [START_REF] Jaoul | Measurement based accurate definition of the SOA edges for SiGe HBTs[END_REF], NPN SiGe HBTs [START_REF] Chevalier | A 55 nm triple gate oxide 9 metal layers SiGe BiCMOS technology featuring 320 GHz fT / 370 GHz fMAX HBT and High-Q Millimeter-Wave Passives[END_REF] based on an advanced BICMOS 55nm technology from ST, as well as on the SG13S technology from IHP [START_REF] Rücker | A 0 . 13µm SiGe BiCMOS Technology Featuring fT/fmax of 240/330 GHz and Gate Delays Below 3 ps[END_REF]. Note that, even though a particular type of stress condition was validated for all the different technologies, with the aging tests conducted at different times, only selected results from each are shown, simply to highlight the versatility of the HICUM AL V3.01 across technologies and aging conditions and not to intent a 1-to-1 comparison between the different technologies. Fig. 7 shows the evolution of IREpS as a function of stress duration for the three technologies under study and corresponding HICUM AL V3.01 simulations, depicting very good accuracy and versatility of the model. 8 (a) shows the results for HBTs from IHP under a specific DC mixed-mode stress condition of IE,stress of 0.12 mA and a VCB,stress of 3V. Both VCB and IE were programmed to switch in an interval of 360s, between the initial (VCB,stress of 3V and IE,stress of 0.12 mA) and a high-current recovery phase (VCB,stress of 0V and a IE,stress of 5.2 mA at 3 times the current at peak-fT). Significant recovery can be observed during the high-current phase (VCB,stress = 0 V), owing to the temperature-annealing effect during this phase. In contrast, for the results depicted in Fig. 8 (b) for ST technologies, a series of stress conditions were used that include VCB,stress bias switching between high and low states (without triggering the thermal/ annealing phase). The results show the evolution of the HiCuM parameter IrEpS, an indicator of the trap generation at the E-B spacer oxide interface, depicting little to no recovery when the stress bias is lowered. This affirms that without annealing or some sort of temperature treatment (as done for Fig. 8(a)) HC induced damage do not show any recovery in SiGe HBTs. Due to the difference in spacer morphology and emitter dimension for both of these technologies, slightly different dynamics of trap generation and H-diffusion have been observed. However, despite the mismatch in current densities, geometries and Hdiffusion scenarios, recovery is not observed for either technology without an annealing cycle.

C. Circuit Aging Simulation

To evaluate the electrical performances of four current mirror architectures (Fig. 9) under mixed-mode operating conditions circuit simulations have been performed for a maximum stress duration of 1000 h using HICUM AL V3.01 for a transient simulation time of 10 µs (with an accelerating factor, ATSF of 3.6×10 11 ) [START_REF] Mukherjee | Hot Carrier Degradation in SiGeHBTs: A Physical and Versatile Aging Compact Model[END_REF][START_REF] Couret | Scaled formulation for trap generation near the EB spacer oxide interface in SiGe HBTs[END_REF]. The following analyses demonstrates how HICUM AL V3.01 can be used in circuit simulation in order to clearly identify the weakest circuit elements, thus allowing the designers to improve the circuit robustness and performances. To assess the robustness of the current mirror configurations in Fig. 9, the output voltage is set to 3 V while the input current is kept constant at a value of 100 µA. The transistor operating points for each current mirror configuration are shown in Fig. 10 (a). It can be observed that the transistor T2 from the simple current mirror operates under strong avalanche conditions, with an equivalent VCB of 2.25 V. Moreover, the output transistors (T4) of the cascode, Wilson and balanced Wilson also exhibit a large VCB of 1.5 V and a VCE of 2.25 V, well beyond the conventional BVCEO. This particular operating condition for the current mirror configurations likely leads to mixed-mode operating stress induced degradation due to the activation of the impact-ionization mechanism for several transistors. From Fig. 10 (b), it can be observed from the IREpSs of different transistors that only four transistors suffer from hot-carrier degradation: the transistor T2 of the simple current mirror and the transistors T4 of the three other configurations. However, even if the transistor is significantly impacted by HCD under mixed-mode stress, circuit electronic functions are not influenced by HCD, as the transfer current ratio does not evolve over the stress duration.

D. Aging Tests on a broadband amplifier

Following the circuit simulation study in the preceding section, HICUM-AL v3.01 has been validated against dynamic aging test results on an actual test circuit, a broadband amplifier [START_REF] López | High speed BiCMOS linear driver core for segmented InP Mach-Zehnder modulators[END_REF], fabricated by IHP Microelectronics. As depicted in Fig. 11, the circuit consists of 11 high-speed SiGe HBT from IHP's SG13S technology, with differential inputs (InP and InN) and outputs (OutP and OutN), along with three DC supplies, ICM (common mode), VCCi (input), and VCCo (output). Finally, as inferred by the analysis in section C that output stages are principally affected by HCD, the aging flag was turned on for T9 only. The corresponding simulation result is depicted by the green curve in Fig. 12 (c) where a 60 μA reduction is observed in the output supply current, close to the case where all transistors are aged. This indicates that the main contribution for the variation of ICCo comes from the impact of hot-carrier degradation on T9 which leads to a reduction of its current gain [START_REF] Couret | Impact of SiGe HBT hot-carrier degradation on the broadband amplifier output supply current[END_REF].

IV. CONCLUSION

The search for high output power density in the SiGe HBT power amplifier leads to operate the transistor at high collectoremitter voltage, often close or beyond the save operating-area.

Biasing state-of-the-art SiGe HBTs at peak fT or beyond provokes noticeable self-heating and the SOA is thermally limited. Under pulsed operation (DC and AC), these limits can be exceeded, but small-signal parameters are very sensitive to the applied pulse width. From a technological point of view, we observed that deep trench isolation has a huge negative impact at high current densities and medium to high VCE. The resulting device degradation when operating the transistor at high IC and/or VCE does not necessarily entail a degradation of circuit performance. A study of different current mirror configurations has shown that despite the fact of device degradation due to HCD, the circuit electronic functions are not altered. However, a similar study on a broadband amplifier shows a significant evolution of the output supply current whereas the differential gain is not affected by the aging.

Fig. 1 :

 1 Fig. 1: Cross-section of a) EEB architecture SG13G3, b) the 55-nm Si/SiGe BiCMOS architecture

Fig. 2 :

 2 Fig. 2: Output characteristics for VBE = 0.85, 0.9 and 0.95 V measured with various pulse widths from 150 ns up to 1.2 μs for an HBT with (a) AE = 8*0.11 × 1.0 μm 2 , and (b) AE = 0.1 × 4.42 μm 2 , duty cycle DC is 10%

Fig. 3 .

 3 Fig. 3. Pulsed measured AC data, (a) S11-IHP, (b) S11-ST, (c) h21-IHP, (d) h21-ST, (e) Mason gain-IHP, (f) Mason gain-ST for an DC pulse width of 150ns and an RF pulse width of 50ns. The applied VBE voltage is 0.9V and VCE varies: VCE=1.0V (red), 1.5V (blue) and 2.0V (cyan), frequency varies from 1 to 40GHz, duty cycle DC is 10%, duty cycle RF is 3%

Fig. 4 .

 4 Fig. 4. Pulsed measured AC data, (a) S11-IHP, (b) S11-ST, (c) h21-IHP, (d) h21-ST for VBE=0.9V and VCE=2V for different pulse width, duty cycle DC is 10%, duty cycle RF is 3%

Fig. 5 :

 5 Fig. 5: (a) Schematic device cross-section showing the physical location of HCD, (b) H-diffusion scenario at the E-B spacer oxide interface.

Fig. 6 :

 6 Fig. 6: Organization of the (a) R-D, (b) OT and (c) Diffusion modules of HICUM AL V3.01.

Fig. 7 :

 7 Fig. 7: Comparison between aging tests and HICUM-AL-V3.01 simulation: Evolution of excess base current at VBE= 0.7 V for HBT technologies from (a)Infineon[START_REF] Jacquet | Reliability of high-speed SiGe: C HBT under electrical stress close to the SOA limit[END_REF], (b) IHP[START_REF] Mukherjee | A physical and versatile aging compact model for hot carrier degradationin SiGe HBTs under dynamic operating conditions[END_REF] and (c) ST[START_REF] Chevalier | A 55 nm triple gate oxide 9 metal layers SiGe BiCMOS technology featuring 320 GHz fT / 370 GHz fMAX HBT and High-Q Millimeter-Wave Passives[END_REF].Next the model simulations were validated against dynamic mixed-mode aging tests conducted on two of the technologies under test in order to affirm time-invariance of the aging

Fig. 8 :

 8 Fig.8: Evolution of excess base current under dynamic mixed-mode stress conditions comparing the measurements (symbols) and HICUM-AL V3.01 (line) for the technologies from (a) IHP[START_REF] Rücker | A 0 . 13µm SiGe BiCMOS Technology Featuring fT/fmax of 240/330 GHz and Gate Delays Below 3 ps[END_REF] and (b) ST[START_REF] Chevalier | A 55 nm triple gate oxide 9 metal layers SiGe BiCMOS technology featuring 320 GHz fT / 370 GHz fMAX HBT and High-Q Millimeter-Wave Passives[END_REF].

Fig. 9 :

 9 Fig. 9: Schematic of (a) the simple, (b) the cascode, (c) the Wilson and (d) the balanced Wilson current mirrors.

Fig. 10 :

 10 Fig. 10: (a) IC -VCE characteristics highlighting the operating conditions for each current mirror configuration and (b) evolution of IREpS as a function of stress duration for different transistors of the current mirror configurations at IIN = 100 µA, VOUT = 3 V.

Fig. 11 :

 11 Fig. 11: Schematic of the broadband amplifier under test.

Fig. 12 :

 12 Fig. 12: (a) JC -VCE characteristics highlighting the operating conditions of different transistors, (b) evolution of differential gain after 50 h of aging test, (c) evolution of output supply current showing measurement (symbols) and simulations (lines), (d) evolution of model-extracted IREpS for the 11 transistors.A dynamic aging stress test is performed on the test structure by applying an input RF power of -15 dBm at ICM = 3 V, VCCi = 3.3 V, VCCo = 3.7 V and TAMB = 25°C. The corresponding operating conditions of the 11 transistors are summarized in Fig.12 (a). Note that, all transistors, excepted T10 and T11, are biased below their BVCEOs, so as not to activate HCD for the other transistors, whereas both self-heating and weak avalanche may occur. On the other hand, a larger degradation of T10 and T11 is expected due to their biasing beyond BVCE0. As observed from Fig.12 (b), the differential gain Sdd21, exhibits negligible variation (0.06 dB) after 50 hours of RF stress, which can be attributed to equipment calibration shift over time. Thus, the circuit figures of merit are not altered by HCD. However, a degradation of the output supply current ICCo of 100 μA has been measured (symbols in fig.12 (c)), after 50 hours of RF stress. To identify the source of this degradation, Fig.12 (d)shows the evolution of the IREpS for different transistors. To complement this analysis, first, the black curve in Fig.12 (c) was obtained by turning the aging flags on for all transistors that shows a total of 80 μA reduction in the output supply current compared that closely matches the results obtained from aging tests (symbols), thus validating the accuracy of the aging compact model on circuit level. Next, in order to systematically locate the circuit elements most susceptible to hot-carrier degradation, the aging flag was only activated for T10 and T11,


J Klinger

R Voituriez

O Bénichou

Splitting Probabilities of Jump Processes

We derive a universal, exact asymptotic form of the splitting probability for symmetric continuous jump processes, which quantifies the probability π0,x(x0) that the process crosses x before 0 starting from a given position x0 ∈ [0, x] in the regime x0

x. This analysis provides in particular a fully explicit determination of the transmission probability (x0 = 0), in striking contrast with the trivial prediction π0,x(0) = 0 obtained by taking the continuous limit of the process, which reveals the importance of the microscopic properties of the dynamics. These results are illustrated with paradigmatic models of jump processes with applications to light scattering in heterogeneous media in realistic 3d slab geometries. In this context, our explicit predictions of the transmission probability, which can be directly measured experimentally, provide a quantitative characterization of the effective random process describing light scattering in the medium.

The splitting probability quantifies the likelihood of a specific outcome out of several alternative possibilities for a random process [START_REF] Redner | A Guide to First-Passage Processes[END_REF][START_REF] Van Kampen | Stochastic Processes in Physics and Chemistry[END_REF][START_REF] Hughes | Random Walks and Random Environments[END_REF][START_REF] Gardiner | Handbook of Stochastic Methods for Physics[END_REF]. While these quantities can be defined for general d-dimensional stochastic processes and any number of possible outcomes [START_REF] Condamin | Proceedings of the National Academy of Sciences[END_REF][START_REF] Chevalier | [END_REF], most examples of applications concern 1-dimensional processes with two outcomes; one then defines π 0,x (x 0 ) as the probability that the process crosses x before 0 starting from x 0 . A celebrated example is given by the Gambler's ruin problem [START_REF] Redner | A Guide to First-Passage Processes[END_REF], schematically quantified by the splitting probability that a 1-dimensional random walker (figuring the gambler's fortune) reaches 0 (complete ruin) before a fixed given threshold; other examples are given by the fixation probability of a mutant in the context of population dynamics [START_REF] Moran | The Statistical processes of of evolutionary theory[END_REF], or the melting probability of a heteropolymer [START_REF] Oshanin | [END_REF], which can be re-expressed in terms of splitting probabilities. A key example, to which we will refer through this paper, is given by the transmission probability of particles (eg photons or neutrons) through a slab of a scattering medium, which has important applications in various fields [9][10][11][12][13]; in this case the transmission probability is nothing but the splitting probability for the particle to reach the exit side rather than being back-scattered.

There is to date no explicit determination of the splitting probability for general jump processes [START_REF] Van Kampen | Stochastic Processes in Physics and Chemistry[END_REF]14,[START_REF] Majumdar | International Summer School on Fundamental Problems in Statistical Physics[END_REF]. Jump processes are defined as follows for d = 1: at each discrete time step n, the walker performs a jump of extension l ∈ R drawn according to a distribution f (l) whose Fourier Transform will be denoted f (k) = ∞ -∞ e ikl f (l)dl. For jump processes, the splitting probability is known to satisfy the following integral backward equation [START_REF] Van Kampen | Stochastic Processes in Physics and Chemistry[END_REF]:

π 0,x (x 0 ) = ∞ x-x0 dx f (x ) + x-x0 -x0
dx π 0,x (x 0 + x )f (x ) [START_REF] Redner | A Guide to First-Passage Processes[END_REF] which results from a partition over the first jump. Even if this equation is linear, there is to date no available so- The jump process is now evolving in a 3 dimensionnal space and is bound to stay between two hyperplanes H1 and H2 distant of x. We now want to evaluate the probability of crossing H2 before H1, starting from x0.

lution with the exception of the exponential distribution f (l) = e -|l|/γ /(2γ); the main difficulty lies in the finite integration range, which prevents the use of classical integral transforms [START_REF] Majumdar | [END_REF].

An important simplification of the problem is achieved by taking a continuous limit. For symmetric jump processes, considered in what follows, the small k expansion of f (k) reads

f (k) = k→0 1 -(a µ |k|) µ + o(k µ ) (2)
where a µ defines the microscopic characteristic length scale of the process. Two limit behaviors emerge [17,18]. For µ = 2 the variance of the jump distribution is finite and the process is known to converge at large times to Brownian motion; for 0 < µ < 2, the process converges instead to an α-stable Levy process of parameter µ. Hence, there are three independent length scales in the problem: x 0 , x, a µ , which can lead to two distinct asymptotic regimes. Taking a µ x 0 < x defines the continuous limit of the problem (1), whose solution can be obtained and reads [19][20][21]:

π 0,x (x 0 ) = Γ(µ) Γ 2 ( µ 2 ) x 0 x 0 [u(1 -u)] µ/2-1 du. (3)
The regime x 0 x is of particular interest and has received a marked attention [21]. One obtains from (3) that this regime is given by [START_REF]Remarkably, this scaling behavior with x0/x has been generalized to non Markovian scale invariant processes[END_REF] :

π 0,x (x 0 ) ∼ aµ x0 x 2Γ(µ) µΓ 2 ( µ 2 ) x 0 x µ 2 . ( 4
)
As explained above, a key application of splitting probabilities is the determination of the transmission probability of particles through a slab, that can be defined as π 0,x (x 0 = 0). The blunt use of the continuous limit (4) yields π 0,x (x 0 = 0) = 0, in clear contradiction with the expected result for a jump process with finite microscopic length scale a µ , for which π 0,x (x

0 = 0) > ∞ x dx f (x ) > 0.
Finally, the determination of the transmission probability requires to consider the second, distinct regime x 0 a µ and thus to go beyond the continuous limit (4); this is the main purpose of this letter.

Jump processes with finite microscopic length scale a µ have proved to be relevant in various contexts [START_REF] Ziff | [END_REF]. They provide emblematic models of transport of photons or neutrons in scattering media [9]. More recently, they have gained renewed interest in the context of selfpropelled particles, be them artificial or living, such as active colloids, cells or larger scale animals [14,[24][25][26][27]. In what follows, we derive a universal form for the splitting probability for continuous jump processes of finite length scale a µ in the regime x 0 a µ x, which provides in particular an explicit determination of the transmission probability (x 0 = 0), and reveals the importance of the microscopic properties of the process. These results are illustrated with paradigmatic models of jump processes with applications to light scattering in heterogeneous media.

General results. We first derive an asymptotic expression of the splitting probability π 0,x (x 0 ) for general 1d continuous symmetric jump processes of characteristic microscopic length scale a µ as defined above in the limit x → ∞. Denoting F 0,x (n|x 0 ) the probability that the process starting from x 0 ∈ [0, x] crosses 0 before x > 0 for the first time after exactly n steps, and making a partition over the crossing time yields :

1 -π 0,x (x 0 ) ≡ π 0,x (x 0 ) = ∞ n=1 F 0,x (n|x 0 ). (5)
This exact equation expresses the splitting probability in terms of two targets first-passage time distributions F 0,x (n|x 0 ), for which no explicit solutions are available for general jump processes. Adapting the approach introduced for scale invariant processes in 1d [21] and then extended to d-dimensional compact cases [28], we next show that in the asymptotic limit x → ∞, the splitting probability of jump processes can in fact be re-expressed in terms of one target first-passage time distributions. We first note that in (5) the right hand side involves trajectories that cross 0 before x; most of these events thus occur within the typical number of steps n typ needed to cross x. In the regime x a µ , x 0 , we argue that n typ is simply the timescale to cover a distance x [17] and thus satisfies n typ ∼ αx µ where α is a process dependent constant (independent of x 0 ). We next remark that for time scales n < n typ , the target at x is irrelevant so that F 0,x (n|x 0 ) F 0,∞ (n|x 0 ), which leads to

π 0,x (x 0 ) ∼ ntyp n=1 F 0,∞ (n|x 0 ) ≡ 1 -q(x 0 , n typ ) (6)
where q(x 0 , n) = ∞ k=n+1 F 0,∞ (k|x 0 ) is the survival probability, ie the probability that the process never crosses 0 during its n first steps, and F 0,∞ (k|x 0 ) is the probability of crossing 0 after exactly steps. We next make use of the asymptotic behavior of q(x 0 , n) obtained in [29], which yields for 1 (x 0 /a µ ) µ n :

q(x 0 , n) ∼ 1 √ n a -µ 2 µ √ πΓ(1 + µ 2 ) x µ 2 0 . (7)
Combining ( 4) and ( 7) finally yields the coefficient α ∼ n typ /x µ defined above, and thus the following determination of n typ , valid for any x 0 x:

n typ ∼ 2 µ-1 Γ( 1 + µ 2 ) -2 x a µ µ . (8)
In order to determine the dependence on x 0 of the splitting probability, we use next the large n behavior of the survival probability given by [29]:

q(x 0 , n) ∼ n→∞ 1 √ n 1 √ π + V (x 0 ) , (9)
where V (x 0 ) is defined by its Laplace transform:

LV (λ) = ∞ 0 V (x 0 )e -λx0 dx 0 = 1 λ √ π exp - λ π ∞ 0 dk λ 2 + k 2 ln(1 -f (k)) -1 , (10)
and f (k) is the Fourier transformed jump distribution defined above. Using equation ( 6) and the above given asymptotic behavior of n typ (8), we finally obtain the following general explicit asymptotic determination of the splitting probability of jump processes :

lim x→∞ π 0,x (x 0 ) A µ (x) = 1 √ π + V (x 0 ) (11)
where

A µ (x) = a µ x µ/2 2 µ-1 Γ 1 + µ 2 . ( 12
)
This holds for any fixed x 0 , including the regime x 0 a µ that we intended to determine. This result thus elucidates the dependence of the splitting probability on x (in the regime x x 0 , a µ ), and, up to Laplace inversion, on x 0 . In particular, the asymptotic behavior for x 0 a µ can be derived explicitly and yields:

V (x 0 ) =                                      -π -3 2 ∞ 0 dk log(1 -f (k)) x 0 + o(x 0 ) if f (k) = k→∞ o(k -1 ) β 2 √ πΓ(1+ν) cos(πν/2) x ν 0 + o(x ν 0 ) if f (k) ∼ k→∞ βk -ν with ν < 1 -β π 3/2 x 0 ln(x 0 ) + o (x 0 ln(x 0 )) if f (k) ∼ k→∞ βk -1 (13)
Of note, the linear dependence of the auxiliary function 13) was given in [29]. Interestingly, we find that the scaling of the splitting probability with x 0 a µ is not universal and can be sublinear depending solely on the small scale behavior of the jump distribution f (l); in particular it is independent of the large scale behavior of f (l), and thus of µ.

V (x 0 ) on x 0 obtained for f (k) = k→∞ o(k -1 ) in (
Remarkably, although V (x 0 ) and thus π 0,x (x 0 ) (see (11)) generically depend on the jump process through the full jump distribution f (l), the asymptotic transmission probability π 0,x (0) in fact depends on the jump distribution only trough µ and a µ and takes the simple, explicit form:

π 0,x (0) ∼ x→∞ 2 µ-1 √ π Γ 1 + µ 2 a µ x µ 2 . ( 14
)
Even though the above derivation involves the uncontrolled asymptotics (6), we claim that our main results (11) and ( 14) are exact; below we confirm these results either analytically or numerically on representative examples of jump processes.

Jump processes with finite second moment. We start by considering continuous jump processes with a finite second moment, corresponding to the case µ = 2 in (2), which we illustrate by the class of Gamma jump processes of order n > -1, whose jump distributions read

f (l) = 1 2γ n+1 Γ(n + 1) |l| n e -|l| γ , (15)
so that a 2 = γ (n + 1)(n + 2)/2. For n = 0, this corresponds to the classical exponential jump distribution f (l) = e -|l|/γ /(2γ), for which, as mentioned above, the splitting probability is known exactly for all values of parameters [START_REF] Van Kampen | Stochastic Processes in Physics and Chemistry[END_REF], and satisfies in the regime x 0 , a 2 x:

π 0,x (x 0 ) ∼ x→∞ γ x 1 + x 0 γ . ( 16
)
Calculating V (x 0 ) from ( 10), one verifies explicitly the agreement of this exact result with (11). Note that in this example

f (k) = k→∞ o(k -1
), so that one verifies in the x 0 a 2 regime the linear dependence on x 0 predicted by (13) (with the correct prefactor, see SM).

For n = 1, one obtains the so-called Gamma jump process defined by the jump distribution f (l) = 1 2γ 2 |l|e -|l|/γ . To the best of our knowledge the splitting probability for this jump process is not known; it can be obtained explicitly for all values of parameters as we proceed to show. Let us denote by D the differential operator, and check that the following identity holds:

D 2 - 1 γ 2 2 f (y) = 1 γ 4 δ(y) + 1 γ 2 δ (2) (y) (17)
where δ (2) is the second derivative of the Dirac delta function. Applying the operator (D 2γ -2 ) 2 to equation ( 1) yields (where derivatives are taken with respect to x 0 ) :

D 2 - 1 γ 2 2 π 0,x (x 0 ) = 1 γ 4 π 0,x (x 0 ) + 1 γ 2 D 2 π 0,x (x 0 ) (18
) and thus

D 4 π 0,x (x 0 ) - 3 γ 2 D 2 π 0,x (x 0 ) = 0. ( 19
)
The splitting probability is then obtained as:

π 0,x (x 0 ) = Ae - √ 3 γ x0 + Be + √ 3 γ x0 + Cx 0 + E (20)
where A, B, C, E are determined by using (1). This provides finally an explicit, exact determination of the splitting probability (see SM for explicit expressions) for all values of the parameters for the Gamma jump process. Calculating V (x 0 ) from ( 10), one verifies explicitly the agreement of this exact result for all x 0 x with (11) (see SM). In particular, in the x 0 a 2 x regime, the splitting probability satisfies:

π 0,x (x 0 ) ∼ x→∞ 1 x √ 3γ + (2 √ 3 -3)x 0 + o(x 0 ) . ( 21
)
This linear scaling with x 0 is in agreement with equation (13) (with the correct prefactor), as expected since

f (k) = k→∞ o(k -1
). Finally, these two examples for n = 0, 1 provide analytical validations supporting the exactness of our results (11) and (14). Additionally, we show in SM that the asymptotic splitting probability for higher or lower order Gamma jump processes can be derived explicitly, and is confirmed by numerical simulations for n = 2 and n = -1/2 in Fig. 2. 11) and ( 13): for n = -1/2, one has ν = 1/2 and a sublinear dependence on x0, while for n = 2, one has a linear dependence on x0. Theoretical predictions (dashed lines) are obtained by numerical inverse Laplace Transform of (10), while simulations (squares) are averaged over 10 6 trials.

Levy Flights. For jump processes with infinite second moment ie µ < 2 in (2) -called Levy flights [17,18,30,31], no exact results for the splitting probability are available for generic a µ , x 0 . We thus resort to numerical simulations to validate predictions (11) to (14) (see Fig. 3). First, the prediction ( 14) of the transmission probability is confirmed and in particular fully captures the dependence on x (including the prefactor) that is controlled by the large scale behavior of f (l), parameterized by µ and a µ only. In turn, (11) captures the dependence on x 0 , which can lead to different scalings depending on the l → 0 behavior of the jump distribution f (l). The linear dependence on x 0 is illustrated by the α-stable jump distribution of parameter µ defined by f (k) = e -(aµ|k|) µ , which verifies f (k) = k→∞ o(k -1 ); an example of sublinear scaling with x 0 is provided by the jump distribution f (l)

∝ 1 √ |l|(1+|l|)
, which corresponds to ν = 1/2 in (13) and has an infinite second moment (µ = 1/2). Our results are thus also validated in the case of jump processes with infinite second moment.

Application to effective 1D problems. In this section we show how our formalism applies to higher dimensional jump processes evolving between two parallel hyperplanes H 1 and H 2 ; coming back to our initial example of the transmission of particles (eg photons or neutrons) through a slab of a scattering medium, the case d = 3 is of particular interest. The trajectory is then naturally described as a 3d jump process, where at each step, the direction of the jump is drawn uniformly on the unit sphere and its length r is drawn according to a distribution p(r); typically experiments show that exponential or Levy distributions p(r) are observed, and provide as readout the transmission probability through the exit plane H 2 rather than H 1 . Even if the problem and the behavior is sublinear. For the Levy Flight, x is fixed to 2.10 5 and one finds a linear behavior. Theoretical predictions (dashed lines) are obtained by numerical inverse Laplace Transform of (10), while simulations (squares) are averaged over 10 6 trials.

is 3-dimensional, the determination of the transmission probability amounts to solving for the splitting probability of a 1-dimensional problem, with the effective jump distribution

f (l) = 1 2 ∞ |l| p(r)
r dr [27]. The above formalism is thus directly applicable and provides explicit determinations of the asymptotic splitting probability and in particular of the transmission probability (see SM). In the case of an exponential jump distribution p(r) = 1 γ e -r γ , relevant to classical diffusive media [9], we obtain f (l) = 1 2γ Γ 0, |l| γ , where Γ(x, y) stands for the incomplete Gamma function, yielding f (k) = arctan(kγ) kγ after Fourier transform. Equation ( 11) then providesup to Laplace inversion -the asymptotic expression (for x → ∞) of the splitting probability for any x 0 . In particular (11) and ( 13) yield for x 0 a 2 ≡ γ/ √ 3:

π 0,x (x 0 ) ∼ x→∞ 1 √ 3x γ - x 0 ln(x 0 ) 2 + o (x 0 ln(x 0 )) . (22)
In the case of α-stable jump distributions, which have been shown recently to be relevant to photon scattering in hot atomic vapors [12,13], we obtain f (k) = Γ(µ -1 )-Γ(µ -1 ,(aµk) µ ) aµµk (see SM). As above, this provides the asymptotic expression (for x → ∞) of the splitting probability for any x 0 thanks to (11), and making use of (13) one obtains for x 0 a µ :

π 0,x (x 0 ) ∼ x→∞ Γ 1+µ 2 2 µ-1 (1 + µ) a µ x µ 2 × × 1 √ π - Γ µ -1 a µ µπ 3 2 x 0 ln(x 0 )(1 + o(1)) . (23)
Agreement with simulations in both cases is displayed in Fig. 4. 11) and ( 13). (b) Small x0 behavior of the splitting probability. For both processes, x is fixed to 10 3 and the behavior is sublinear as predicted. Theoretical predictions (dashed lines) are obtained by numerical inverse Laplace Transform of (10), while simulations (squares) are averaged over 10 6 trials.

Conclusion.

We have derived a universal exact asymptotic form for the splitting probability for continuous symmetric processes characterized by a finite length scale a µ which have proved relevant various contexts, such as transport of photons or neutrons in scattering media. This analysis covers the regime x 0 a µ x and provides in particular a fully explicit determination of the transmission probability (x 0 = 0), in striking contrast with the trivial prediction π 0,x (x 0 ) = 0 obtained by taking the continuous limit of the process. This reveals the importance of the microscopic properties of the dynamics. These results are illustrated with paradigmatic models of jump processes with applications to light scattering in heterogeneous media in realistic 3d slab geometries. In this context, our explicit predictions of the transmission probability [START_REF] Ziff | [END_REF], which can be directly measured experimentally, provides in principle a quantitative determination of not only the Levy exponent µ, as already proposed in [12,13], but also of the microscopic length scale a µ . This significantly refines the characterization of the effective random process describing light scattering in the medium.

B. f (k) ∼ k→∞ βk -ν with ν < 1 case
In this case, the integral ∞ 0 ln(1 -f (k))dk is ill-defined. We thus need to take further care in making the large λ expansion. Let us denote A(λ) = LV (λ) + 1 λ √ π . Performing the change of variable k λ = u, one obtains:

A(λ) = 1 λ √ π exp - λ π ∞ 0 dk λ 2 + k 2 ln 1 -f (k) = 1 λ √ π exp - 1 π ∞ 0 du 1 + u 2 ln 1 -f (λu)
and using the large k behavior of f (k):

A(λ) = 1 λ √ π exp β λ ν 1 π ∞ 0 du 1 + u 2 1 u ν + o( 1 λ ν ) = 1 λ √ π exp β λ ν 1 2 cos(πν/2) + o( 1 λ ν ) = 1 λ √ π 1 + β λ ν 1 2 cos(πν/2) + o( 1 λ ν ) = 1 λ √ π + β λ 1+ν 1 2 cos(πν/2) √ π + o( 1 λ 1+ν ) (S5)
Finally, the inverse Laplace Transform yields the following small x 0 behavior :

V (x 0 ) = x0→0 β 2Γ(1 + ν) cos(πν/2) √ π x ν 0 + o(x ν 0 ) (S6) C. f (k) ∼ k→∞ βk -1 case
In this case, the change of variable used above cannot be directly applied since the integral

∞ 0 du 1+u 2 1
u is ill-defined. Let us denote F (k) a primitive of ln(1 -f (k)). Its large k behavior is dictated by that of f and reads : F (k) ∼ -β ln(k). Let us now extract the dominating large λ behavior of A(λ) :

A(λ) = 1 λ √ π exp - 1 π ∞ 0 du 1 + u 2 ln 1 -f (λu) = 1 λ √ π exp - F (λu) πλ(1 + u 2 ) ∞ 0 - 1 πλ ∞ 0 2udu (1 + u 2 ) 2 F (λu) = λ→∞ 1 λ √ π exp A λ - 1 πλ ∞ 0 2udu (1 + u 2 ) 2 β ln(λu) + o ln(λ) λ with A some irrelevant constant A(λ) = λ→∞ 1 λ √ π exp -β ln(λ) πλ + o ln(λ) λ = λ→∞ 1 λ √ π - β π 3 2 ln(λ) λ 2 + o ln(λ) λ 2 (S7)
Finally, the inverse Laplace Transform yields the following small x 0 behavior :

V (x 0 ) = x0→0 - β π 3 2
x 0 ln(x 0 ) + o (x 0 ln(x 0 )) (S8)

II. SPLITTING PROBABILITY OF THE LAPLACE JUMP PROCESS

In this section, we show that the splitting probability of the Laplace jump process defined by its jump distribution f (l) = 1 2γ e -|l| γ satisfies equation (11) of the main text.

A. Exact determination

The exact calculation can be found in 1 . We here state the result for completeness :

π 0,x (x 0 ) = x 0 + γ x + 2γ (S9)
B. Large x limit and matching with V (x0)

On the one hand, following the main text, the rescaling function reads A 2 (x) = γ √ π

x and thus

lim x→∞ π 0,x (x 0 ) A 2 (x) = 1 √ π + x 0 √ πγ (S10)
On the other hand, for the Laplace jump process, it is shown in 2 equation (72) that :

V (x 0 ) = x 0 √ πγ (S11)
yielding the expected result.

C. Small x0 behavior

For the Laplace jump process, the Fourier Transform reads f (k) = 1 1+γ 2 k 2 . We note that for small x 0 , V (x 0 ) is linear in x 0 and

-π -3 2 ∞ 0 dk log γ 2 k 2 1 + γ 2 k 2 = 1 √ πγ (S12)
thus confirming the expression of the prefactor.

III. SPLITTING PROBABILITY OF THE GAMMA JUMP PROCESS

In this section we pursue the same goal as in the previous section for the Gamma Jump Process, defined by its jump distribution f (l) = 1 2γ 2 |l|e -|l| γ .

A. Exact determination

Let us provide the exact expression of the splitting probability for the Gamma jump process. Following the main text, recall that the splitting probability obeys the following integral equation :

π 0,x (x 0 ) = ∞ x-x0 dx f (x ) + x-x0 -x0 dx π 0,x (x 0 + x )f (x ) (S13)
and the following differential equation :

d 4 d 4 x 0 π 0,x (x 0 ) - 3 γ 2 d 2 d 2 x 0 π 0,x (x 0 ) = 0 (S14)
The splitting probability can thus be written as a linear combination of orthogonal solutions :

π 0,x (x 0 ) = Ae - √ 3 γ x0 + Be + √ 3 γ x0 + Cx 0 + E (S15)
with unknown coefficients A, B, C and E depending on x. Injecting this form into the integral equation, we obtain 4 independent equations and determine the unknown coefficients and thus the splitting probability:

A = 2γe √ 3x γ 6 √ 3γ -8γ + 2 √ 3xe √ 3x γ + 3xe √ 3x γ + 6 √ 3γe √ 3x γ + 8γe √ 3x γ + 2 √ 3x -3x B = -90 √ 3γ -156γ 108 √ 3γ + 186γ + 291 √ 3xe √ 3x γ + 504xe √ 3x γ + 828 √ 3γe √ 3x γ + 1434γe √ 3x γ + 21 √ 3x + 36x C = -189 √ 3γ + 327γ + 6 √ 3xe 2 √ 3x γ + 9xe 2 √ 3x γ -9 √ 3γe √ 3x γ + 15γe √ 3x γ + 21 √ 3γe 2 √ 3x γ + 33γe 2 √ 3x γ + 33 √ 3γe 3 √ 3x γ + 57γe 3 √ 3x γ -78 √ 3x + 135x × -538 √ 3γ 2 + 930γ 2 + 6 √ 3x 2 e 2 √ 3x γ + 9x 2 e 2 √ 3x γ -78 √ 3x 2 + 135x 2 -54 √ 3γ 2 e √ 3x γ + 102γ 2 e √ 3x γ + 66 √ 3γ 2 e 2 √ 3x γ + 102γ 2 e 2 √ 3x γ + 94 √ 3γ 2 e 3 √ 3x γ + 162γ 2 e 3 √ 3x γ -21 √ 3γxe √ 3x γ + 39γxe √ 3x γ + 39 √ 3γxe 2 √ 3x γ + 57γxe 2 √ 3x γ + 33 √ 3γxe 3 √ 3x γ + 57γxe 3 √ 3x γ -411 √ 3γx + 711γx -1 E = -66 √ 3γ 2 -66γ 2 + 324 √ 3γ 2 e √ 3x γ + 564γ 2 e √ 3x γ + 5574 √ 3γ 2 e 2 √ 3x γ + 9654γ 2 e 2 √ 3x γ + 114 √ 3γxe √ 3x γ + 198γxe √ 3x γ + 1959 √ 3γxe 2 √ 3x γ + 3393γxe 2 √ 3x γ -21 √ 3γx -27γx × -132 √ 3γ 2 -132γ 2 + 1377 √ 3x 2 e 2 √ 3x γ + 2385x 2 e 2 √ 3x γ -9 √ 3x 2 -9x 2 + 648 √ 3γ 2 e √ 3x γ + 1128γ 2 e √ 3x γ + 11148 √ 3γ 2 e 2 √ 3x γ + 19308γ 2 e 2 √ 3x γ + 228 √ 3γxe √ 3x γ + 396γxe √ 3x γ + 7836 √ 3γxe 2 √ 3x γ + 13572γxe 2 √ 3x γ -72 √ 3γx -72γx -1 (
S16)

B. Large x limit and matching with V (x0)

Let us now turn to the large x behavior the splitting probability. For each of the above given coefficients, we obtain the equivalent for x → ∞:

A ∼ 2γ 2 √ 3x + 3x B ∼ -90 √ 3γ -156γ 291 √ 3xe √ 3x γ + 504xe √ 3x γ C ∼ 1 x E ∼ 1959 √ 3γ + 3393γ 1377 √ 3x + 2385x (S17)
For this jump process, the rescaling function reads

A 2 (x) = √ 3πγ x
and we thus obtain :

lim x→∞ π 0,x (x 0 ) A 2 (x) = 1 √ π + 1 √ π ( √ 3 -1) 2 3 e - √ 3 γ x0 + x 0 √ 3γ + 2 √ 3 -4 3 (S18)
Finally, for the Gamma jump process, it is shown in 2 , equation (77) that :

V (x 0 ) = 1 √ π ( √ 3 -1) 2 3 e - √ 3 γ x0 + x 0 √ 3γ + 2 √ 3 -4 3 (S19)
yielding the agreement with equation ( 11) of the main text.

C. Small x0 behavior

For the Gamma jump process, the Fourier Transform reads f (k) = 1-k 2 γ 2 (1+γ 2 k 2 ) 2 . We note that for small x 0 , V (x 0 ) is linear in x 0 and

V (x 0 ) = x0→0 2 - √ 3 √ π x 0 γ + o(x 0 ) (S20)
One can show as well that

-π -3 2 ∞ 0 dk log 1 - 1 -k 2 γ 2 (1 + γ 2 k 2 ) 2 = 2 - √ 3 √ π x 0 γ (S21)
thus confirming the expression of the prefactor in the small x 0 limit and agreeing with equation ( 13) of the main text.

IV. FOURIER TRANSFORMED JUMP DISTRIBUTIONS FOR VARIOUS GAMMA JUMP PROCESSES

We recall that the jump distribution reads, for n > -1:

f (l) = 1 2γ n+1 Γ(n + 1)
|l| n e -|l| γ .

(S22)

Let us give the Fourier transforms for the various values of n used in the main text:

• n = 0 f0 (k) = 1 1 + γ 2 k 2 (S23) • n = 2 f2 (k) = 1 -3k 2 γ 2 (1 + γ 2 k 2 ) 3 (S24) • n = -1/2 f-1 2 (k) = 1 + 1 + k 2 γ 2 2 + 2γ 2 k 2 (S25)
From these expressions, one easily obtains the large and small k behavior of the Fourier Transform and thus the transition probabilities and small x 0 asymptotic limit of the splitting probabilities for these Gamma jump processes, as claimed in the main text.

V. EFFECTIVE 1d JUMP DISTRIBUTION IN 3d GEOMETRIES

In this section, we provide the derivation of the effective 1d jump distribution in the 3d slab geometry. Let us first remind that for the jump processes considered, the projection on one coordinate of the jump distribution reads

f (l) = 1 2 ∞ |l| p(r) r dr (S26)
with p(r) the distribution of the module, the angle being distributed uniformly on the unit sphere.

A. Exponential distribution

We first consider the case where the module of the jump is distributed according to p(r) = 1 γ e -r γ . One then has: The large and small k expansions read:

f (l) = 1
f (k) = k→0 1 - γ 2 k 2 3 + o(k 2 ) f (k) = k→∞ π 2γk + o(k -1 ) (S29)

FIG. 1 .

 1 FIG. 1. (a) the one dimensional jump process evolves in the bounded interval [0, x]. One is interested in the probability π0,x(x0) of crossing x before 0 starting from x0 ,as shown in the diagram. (b)The jump process is now evolving in a 3 dimensionnal space and is bound to stay between two hyperplanes H1 and H2 distant of x. We now want to evaluate the probability of crossing H2 before H1, starting from x0.

FIG. 2 .

 2 FIG. 2. (a)Transmission probability for examples of Gamma jump processes. After rescaling according to(14), the transmission probabilities collapse. (b) Small x0 behavior of the splitting probability, as predicted by (11) and (13): for n = -1/2, one has ν = 1/2 and a sublinear dependence on x0, while for n = 2, one has a linear dependence on x0. Theoretical predictions (dashed lines) are obtained by numerical inverse Laplace Transform of (10), while simulations (squares) are averaged over 10 6 trials.

FIG. 3 .

 3 FIG. 3. (a)Transmission probability for a jump process with distribution f (l) = (2π |l|(1 + l)) -1 (denoted F process), yielding µ = 1/2 and ν = 1/2 and a Levy flight with µ = 1 and aµ = 2. The transmission probabilities (including prefactors) are accurately predicted. (b) Small x0 behavior of the splitting probability. For the F process, x is fixed to 10 4 and the behavior is sublinear. For the Levy Flight, x is fixed to 2.10 5 and one finds a linear behavior. Theoretical predictions (dashed lines) are obtained by numerical inverse Laplace Transform of (10), while simulations (squares) are averaged over 10 6 trials.

FIG. 4 . 1 .

 41 FIG. 4. (a) Transmission probability for an exponential jump process with γ = 1, and a Levy flight with µ = 1 and aµ = 1. The transmission probabilities (including prefactors) are accurately predicted by (11) and (13). (b) Small x0 behavior of the splitting probability. For both processes, x is fixed to 10 3 and the behavior is sublinear as predicted. Theoretical predictions (dashed lines) are obtained by numerical inverse Laplace Transform of (10), while simulations (squares) are averaged over 10 6 trials.

Van N.G. Kampen. Stochastic Processes in Physics and Chemistry. Elsevier,

2 Satya N. Majumdar, Philippe Mounaix, and Grégory Schehr. Survival probability of random walks and Lévy flights on a semi-infinite line. Journal of Physics A: Mathematical and Theoretical, 50(46):465002, nov 2017.

This Supplementary Material presents details of calculations supporting the main text. We provide :

• details on the asymptotic behavior of V (x 0 ) in the x 0 a µ limit.

• details on the derivation of the splitting probability for Laplace jump processes.

• details on the derivation of the splitting probability for Gamma jump processes.

• details on the Fourier Transforms of the Gamma jump process family.

• details on the derivation of effective jump distributions in 3d geometries.

In this section we provide the derivation of the asymptotic V (x 0 by its Laplace transform:

with f (k) the Fourier transform of the jump distribution f (l). The small x 0 behavior is given by the corresponding large λ behavior of the Laplace Transform.

Note that this case has already been addressed in 2 . However, we reproduce the derivation for completeness. For

)dk converges and, to first order, one has:

yielding:

and thus the corresponding x 0 behavior:

x and thus the following small x 0 behavior :

as claimed in the main text.

B. α-stable jump process

We now turn to an α-stable jump process with module distributed as :

Let us directly compute the Fourier Transform of the effective 1d distribution:

Let us now go into the complex formalism, and we will take the imaginary part at the end:

Finally :

and the large and small k expansions read:

yielding result [START_REF] Ziff | [END_REF] of the main text.


J Klinger

A Barbier-Chebbah

R Voituriez

O Bénichou

Joint statistics of space and time exploration of 1d random walks

The statistics of first-passage times of random walks to target sites has proved to play a key role in determining the kinetics of space exploration in various contexts . In parallel, the number of distinct sites visited by a random walker and related observables have been introduced to characterize the geometry of space exploration. Here, we address the question of the joint distribution of the firstpassage time to a target and the number of distinct sites visited when the target is reached, which fully quantifies the coupling between kinetics and geometry of search trajectories. Focusing on 1-dimensional systems, we present a general method and derive explicit expressions of this joint distribution for several representative examples of Markovian search processes. In addition, we obtain a general scaling form, which holds also for non Markovian processes and captures the general dependence of the joint distribution on its space and time variables. We argue that the joint distribution has important applications to various problems, such as a conditional form of the Rosenstock trapping model, and the persistence properties of self-interacting random walks.

Quantifying the efficiency of space exploration by random walkers is a key issue involved in a variety of situations. Applications range from reactive particles diffusing in the presence of catalytic sites, living organisms looking for resources, to robots cleaning or demining a given area [START_REF] Hughes | Random walks and random environments[END_REF][START_REF] Bénichou | [END_REF][3]. In this context, two important classes of observables have been considered.

First, the statistics of first-passage times (FPTs) to target sites of interest has proved to play a key role in determining the kinetics of space exploration [START_REF] Redner | A guide to First-Passage Processes[END_REF][START_REF] Metzler | First passage problems: recent advances[END_REF][START_REF] Bray | [END_REF]. The case of first-passage times in confined domains was found to be particularly relevant to assess the efficiency of target search processes, and has lead to an important activity [7][8][9][10]. Related observables, such as the cover time of a domain [11][12][13] or the occupation time of a sub domain have also been considered in this context [14][15][16][START_REF] Condamin | Proceedings of the National Academy of Sciences[END_REF].

A second class of observables has been introduced to characterize the geometry of the territory explored by random walkers. In particular, the number of distinct sites visited (or the so called Wiener sausage in a continuous setting) by a random walker after n step, which quantifies the overall territory swept by the random walker, has been the focus of many studies with a broad range of applications [START_REF] Hughes | Random walks and random environments[END_REF][START_REF] Weiss | Aspects and Applications of the Random Walk[END_REF][START_REF] Larralde | [END_REF][20]. Notable extensions include the number of distinct sites visited by p independent walkers [START_REF] Larralde | [END_REF], the case of fractal geometries [START_REF] Blumen | Optical Spectroscopy of Glasses[END_REF][START_REF] Ben-Avraham | Diffusion and reactions in fractals and disordered systems[END_REF], or the case of random stopping times [START_REF] Dayan | [END_REF][24][25][26].

Even if it is clear that both classes of observables are coupled, so far kinetic and geometric properties of exploration have been mainly discussed independently, with the notable exception of [27]. Qualitatively, the firstpassage time to a target of a generic stochastic process carries information about the territory visited before hitting the target : large values of the first-passage time imply large values of the visited territory. However, the quantitative determination of this coupling is still lacking.

Here, we address the question of the joint distribution of the first-passage time to a target and the number of distinct sites visited when the target is reached, which fully quantifies this coupling and gives access to a refined characterization of search trajectories. To the best of our knowledge, this quantity has never been studied so far. The joint law provides two conditional distributions, which allow to answer quantitatively the following questions : (Q 1 ) What is the territory visited by a random walker knowing that it reached a target (and stopped or exited the domain) after a given time? (Q 2 ) How long does it take a random walker to reach a target knowing that it has visited a given number of distinct sites before? We anticipate that these quantities could have applications in various situations where only partial information -either kinetic or geometric -on trajectories is accessible.

Summary of the results. We tackle this general question in the case of 1-dimensional processes, and determine the joint distribution σ(s, n|s 0 ) of the FPT n at the target site 0 and the number s of distinct sites visited by a random walker starting from s 0 (see Fig. 1(a), where x, t are the continuous counterparts of s, n) [52]. Our approach applies to general (space and time) discrete or continuous random walkers, evolving in a semi infinite or finite domain, and yields fully explicit expressions of σ(s, n|s 0 ) for several representative examples of Markovian processes, such as simple symmetric and biased random walks, persistent random walks [START_REF] Hughes | Random walks and random environments[END_REF][START_REF] Weiss | Aspects and Applications of the Random Walk[END_REF] or resetting random walks [28,29], whose definitions are recalled below. In addition, we derive a general scaling form of σ(s, n|s 0 ) in the large s, n regime, which holds also for non Markovian processes and captures the general dependence on s 0 , s, n. Several applications of these central results are then discussed. First, we determine the efficiency of a schematic catalytic reaction [30] by deriving the probability that a diffusing particle has reacted in a domain with Poisson distributed targets before exiting the domain, knowing the exit time (see Fig. 1(b)). Second, we show that the knowledge of the joint distribution σ(s, n|s 0 ) for simple random walks is actu- ally required for determining first-passage properties of a class of strongly non Markovian processes, namely selfinteracting random walks [31][32][33][34][35]. More precisely, with the help of the joint distribution, we derive exactly the full large time behavior of the FPT density of the socalled self-attracting walk (SATW) [36], which has been studied in the context of random search processes as a prototypical example of processes with long-range memory [37][38][39][40] and has important applications in the theoretical description of the trajectories of living organisms such as cells [41].

Discrete processes. We first consider the case of a general Markovian discrete (in space and time) process, which leaves no holes in its trajectory; in other words, the set of visited sites is assumed to be at all times the finite range s min , s max defined by the min (s min ) and max (s max ) values of the random walker's positions. This last hypothesis will hold for all processes presented in what follows. Denoting by s 0 > 0 the starting site, n the step at which the walker reaches the target 0 for the first time and s the number of distinct sites visited up to this random stopping time, we derive a systematic procedure to obtain the joint law σ(s, n|s 0 ). In turn, this joint law gives immediate access to the conditional probabilities mentioned above, ie (i) the distribution of the number s of distinct sites visited before reaching 0 knowing that the random walker has reached 0 at step n :

G sp (s|n, s 0 ) = σ(s, n|s 0 ) ∞ s =s0 σ(s , n|s 0 ) ≡ σ(s, n|s 0 ) F 0 (n|s 0 ) , (1)
where F 0 (n|s 0 ) is the usual FPT distribution to 0 and (ii) the distribution of the FPT to 0 knowing that s distinct sites have been visited before reaching 0 :

G tm (n|s, s 0 ) = σ(s, n|s 0 ) ∞ n =0 σ(s, n |s 0 ) ≡ σ(s, n|s 0 ) µ 0 (s|s 0 ) , (2)
where µ 0 (s|s 0 ) is the distribution of the maximum s before reaching 0. Let us denote F 0,s (n|s 0 ) the probability that the walker reaches zero for the first time at step n, without ever reaching s, and make a partition over the rightmost site s visited before reaching zero. Because the walker reaches 0 before s, one necessarily has s ∈ s 0 , s -1 , which yields F 0,s (n|s 0 ) = s-1 s =s0 σ(s , n|s 0 ). Note that this relation still holds for non Markovian processes. Equivalently, we obtain the key relation

σ(s, n|s 0 ) = F 0,s+1 (n|s 0 ) -F 0,s (n|s 0 ) ≡ D s F 0,s (n|s 0 ), (3)
which allows one to write the joint law σ explicitly in terms of the quantity F 0,s (n|s 0 ).

We next provide a procedure based on backward equations to derive the probability F 0,s (n|s 0 ) in presence of two absorbing sites 0 and s for a given Markovian stochastic process. In this case, the propagator P (s, n|s 0 ), ie the probability for the walker to be at site s after n steps, obeys the backward equation P (s, n + 1|s 0 ) = L s0 [P (s, n|s 0 )] [START_REF] Hughes | Random walks and random environments[END_REF][START_REF] Kampen | Stochastic Processes in Physics and Chemistry[END_REF], obtained by partitioning over the first step of the walk, where L s0 is a linear operator acting on s 0 . For instance, in the case of a simple random walk, L s0 [P (s, n|s 0 )] = 1 2 P (s, n|s 0 + 1) + 1 2 P (s, n|s 0 -1). It is easily seen that F 0,s (n|s 0 ) obeys the same backward equation for 0 < s 0 < s and, introducing the generating function F0,s (ξ|s 0 ) = ∞ n=0 ξ n F 0,s (n|s 0 ), we obtain: F0,s (ξ|s 0 ) = ξL s0 F0,s (ξ|s 0 ) .

Reminding that both 0 and s are absorbing boundaries, we have that, for any n > 0, F 0,s (n|0 or s) = 0 whereas F 0,s (0|0) = 1 and F 0,s (0|s) = 0. In terms of generating functions, we obtain the following boundary conditions: F0,s (ξ|0) = 1 ; F0,s (ξ|s) = 0.

Eq. ( 4), completed by [START_REF] Metzler | First passage problems: recent advances[END_REF], fully determines F0,s (ξ|s 0 ). Making use of (3), we then derive the generating function of the joint law σ.

As an illustration, we obtain in the case of a simple random walk (see supplementary material (SM))

σ(s, ξ|s 0 ) = r + -r - r s + -r s - r s0 + -r s0 - r s+1 + -r s+1 - ( 6
)
where

r ± = 1 ξ (1 ± 1 -ξ 2 )
. Further illustration is provided in SM, where explicit expressions of σ(s, ξ|s 0 ) are determined for the important examples of biased random walks (for which a step is taken to the right with probability p, and to the left with probability 1 -p), persistent random walks (for which each step is taken identical to the previous one with probability p) [START_REF] Ernst | [END_REF]44] and resetting random walks (for which at each step the walker has a probability λ to jump back to its initial position) [28,29,45,46]. Finally, in each case, a series expansion with respect to ξ gives access to an exact determination of σ(s, n|s 0 ) (see SM for validation by numerical simulations), which constitutes the main result of this section; its physical implications are commented below (see discussion and applications).

Continuous space and time. This method is easily adapted to continuous space and time (x, t) Markovian processes. Defining F 0,x (t|x 0 ) as the probability density to reach 0 before x at time t, the continuous counterpart of Eq. ( 3) reads:

σ(x, t|x 0 ) = D x F 0,x (t|x 0 ) (7)
where here D x is the differential operator with respect to x, and the Laplace transform F0,x (p|x 0 ) = ∞ 0 e -pt F 0,x (t|x 0 )dt satisfies the continuous counterpart of Eq (4), (5) (see SM). As an explicit example, for Brownian diffusion with diffusion coefficient D, it is found that the joint law is given by

σ(x, t|x 0 ) = 2Dπ x 3 ∞ k=1 e -(kπ) 2 Dτ k sin(kπx 0 )× × 2(kπ) 2 Dτ -2 - kπ x0 tan(kπx 0 ) , (8)
where x0 = x0

x and τ = t x 2 . Explicit expressions of σ for other continuous Markov processes (biased diffusion and continuous resetting) are presented in SM. Importantly, it is also shown in SM that our approach can be further extended to the case of continuous space but discrete time processes, also known as jump processes, as well as Markovian processes in confined domains.

General scaling form. Beyond the case of Markovian processes, we now show that the joint law σ assumes a general scaling form for symmetric processes, which holds even in the non Markovian case and elucidates its dependence on the parameters s, s 0 , n. Because we are interested only in the large time and space limit, we adopt a continuous formalism and make use of the variables x, x 0 , t. Extending an approach given in [47,48], we derive below a general scaling form for F 0,x (t|x 0 ), which leads to the asymptotic behavior of σ(x, t, x 0 ).

First, note that walkers reaching x before 0 do not contribute to the probability F 0,x (t|x 0 ). Hence, for times shorter than the typical time T typ ∝ x dw needed to reach x (which defines the walk dimension d w of the process), F 0,x (t|x 0 ) behaves as the first-passage time density F 0 (t|x 0 ) in a semi-infinite domain, with a single target in 0. We now assume that this quantity has an algebraic decay with time for t → ∞, quantified by the persistence exponent θ of the process: F 0 (t|x 0 ) ∼ k(x 0 )t -(θ+1) , where k(x 0 ) ∝ x dwθ 0 for x 0 1 [START_REF] Bray | [END_REF]. Because almost all random walkers have either reached 0 or x at times t x dw , we write

F 0,x (t|x 0 ) ∼ F 0 (t|x 0 )g( t x dw ) ∼ k(x 0 )t -(θ+1) g( t x dw ) (9)
where g is a smooth cut-off function with g(0) = 1 and g(y) vanishes for large y. Finally, with the help of ( 7), we obtain the general scaling form for the joint law in the scaling limit defined by x → ∞, t → ∞ with τ = t/x dw fixed : σ(x, t|x 0 ) ∼ h(x 0 ) x dw(θ+1)+1 f (τ ) (10) where, defining f 1 (τ ) = -d w g (τ )τ -θ and N = ∞ 0 f 1 (τ )dτ , we have h(x 0 ) = k(x 0 )N and f = f 1 /N . In addition, h(x 0 ) ∝ x dwθ 0 for 0 1, and f (τ ) is a normalized process dependent function.

Of note, integrating equation ( 10) over t recovers the distribution of the maximum µ 0 (x|x 0 ) = h(x 0 )x -(dwθ+1) before reaching 0, in agreement with known results [47]. In turn, this provides a simple physical interpretation of f (τ ). Making use of ( 2), we obtain the conditional density G tm (t|x, x 0 ) ∼ 1

x dw f (τ ). Thus, f (τ ) is the density of the rescaled variable τ conditioned by the value of the maximum x. In particular, we stress that f is independent of x 0 .

The general relation (10) is confirmed in Fig 2 by numerical simulations for representative examples of both Markovian processes (simple random walks and Riemann walks, ie discrete space and time Levy flights [START_REF] Hughes | Random walks and random environments[END_REF]), and non Markovian processes (Fractional Brownian Motion [49] and the Random Acceleration Process [50], see SM for definitions). Indeed, we find that the conditional density of the FPT knowing the territory covered, which a priori depends on the variables t, x, x 0 , can in fact be rewritten as the distribution f (τ ) of the single reduced variable τ , as shown by the data collapse in the figure .  Next, thanks to the exact Eq. ( 2), and the exact scaling of the distribution µ 0 of the maximum reminded above [47], this observed scaling of f directly confirms (10).

In the case of diffusive random walks, f (τ ) can be determined explicitly by taking x → ∞ and t → ∞ with τ fixed in Eq. ( 8):

f BM (τ ) = 2Dπ 2 ∞ k=1 e -(kπ) 2 Dτ k 2 2(kπ) 2 Dτ -3 . (11)
Of note, this asymptotic conditional distribution holds for any symmetric Markovian random walk satisfying the central limit theorem.

Similarly (see SM), the other conditional distribution defined in (1) can be written from (10) as G sp (x|t, x 0 ) ∼ 1 t 1/dw φ(χ) where the density of the rescaled variable χ = x/t 1/dw is given in terms of f by :

φ(χ) = χ -dw(θ+1)-1 f (χ -dw ) ∞ 0 u -dw(θ+1)-1 f (u -dw )du . ( 12
)
The agreement of this result with numerical simulations is shown in SM. Discussion. The above results yield both exact expressions of the joint law for Markovian processes, and scaling forms for general non Markovian processes, and have important implications. (i) The joint law, because it gives access to all correlation functions x n t m , fully quantifies the coupling between the kinetics of space exploration and the territory explored by a random walker. This coupling manifests itself in the dependence of σ on the rescaled variable τ = t/x dw . (ii) The joint law yields the conditional distributions G sp (see [START_REF] Hughes | Random walks and random environments[END_REF]) and G tm (see [START_REF] Bénichou | [END_REF]), which provide new insights in the quantification of space exploration, and in particular explicit answers to the questions Q 1 , Q 2 raised in introduction. Below, we further illustrate the importance of the joint law and turn to examples of applications of our results.

Application -Conditional Rosenstock problem. The above results provide as a by product an explicit solution to a conditional version of the celebrated Rosenstock problem [START_REF] Hughes | Random walks and random environments[END_REF]30]. We consider a reactive diffusing particle that enters a 1-dimensional chemical reactor at x 0 and leaves it at 0. The reactor contains Poisson distributed catalytic point-like sites of density ρ, which trigger a reaction upon encounter with the reactive particle (see Fig. 1 (b)). The efficiency of such schematic catalytic reaction can be quantified by the probability P t that the reactive particle has reacted with a catalytic site before exiting the domain, knowing the exit time t. This is readily obtained as

P t = ∞ 0 (1 -e -ρx )G sp (x|t, x 0 )dx. ( 13
)
The determination of P t thus requires G sp , and therefore the joint law. Making use of the general scaling (10), we obtain the large time scaling behaviour :

P t ∼ t→∞ ∞ 0
(1 -e -ρt 1/dw u )φ(u)du; (14) this shows that P t is asymptotically a function of the reduced variable ρt 1/dw only, with P t ∝ ρt 1/dw for ρt 1/dw → 0. Equation ( 14) provides, thanks to (12), an explicit determination of P t for all processes for which σ (and thus f ) is known, and in particular elucidates its dependence on the exit time t from the domain (see Fig. 1(b)).

On the example of Brownian motion, one obtains (for x 2 0 /D t 1/(Dρ 2 )):

P t ∼ √ πρ(Dt) 1 2 . ( 15
)
Application -Self-interacting walkers. Next, we show that the joint law can be needed to obtain the firstpassage time distribution. This is the case of selfinteracting random walks, which are defined generically as random walks whose jump probabilities at time n depend on the full set of visited sites at earlier times n < n. We focus on the example of the 1d self-attracting walk (SATW) [36], which has been studied in the context of random search processes as a prototypical example of process with long-range memory, and has recently proved to be relevant to describe the dynamics of motile cells [41]. At each time step, if both its neighboring sites have already been visited, the random walker hops on either of them with probability 1/2. However, if one of them has never been visited, it is chosen with probability β. Note that this can either be an attractive effect (β < 1/2) or a repulsive one (β > 1/2). Since the dynamics of the walk is completely determined by the location of unvisited sites, the determination of the first-passage time distribution requires the knowledge of all times at which unvisited sites have been discovered. Denoting here F 0,s (n|s 0 ) the probability to reach s before 0 for the first time at step n, knowing that the sites 1, s -1 have already been visited, the generating function of σ(s, n|1) can be written as :

σ(s, ξ|1) = ξ 2 s s =3
F0,s (ξ|s -1) F0,s+1 (ξ|s) (16) Solving for F0,s (ξ|s 0 ) yields an explicit expression of σ (see SM). For large s and n, with τ = n s 2 fixed, this yields σ(s, n|s 0 ) = h(s 0 )s - .

(17) The FPT distribution is finally deduced from σ(s, n|1) and yields the following exact asymptotics (see SM):

F 0 (n|s 0 = 1) ∼ n→∞ Γ( 2 β -1) Γ( 1 2β -1 2 ) 2 -1+β 2β n -1-β 2β -1 . (18)
While the n decay is in agreement with the recent determination of the persistent exponent of the SATW relying on a different approach [51], this formalism based on the joint law gives access to the explicit expression of the prefactor for this strongly non Markovian process.

Conclusion.

We have proposed a general method to derive explicit expressions of the joint distribution of the first-passage time to a target and the number of distinct sites visited when the target is reached for 1d random walks. This method yields explicit expressions for several representative examples of Markovian search processes. Furthermore, we showed that the dependence of the joint distribution on its space and time variables is captured by a general scaling form, which holds even for non Markovian processes. We argue that the joint distribution could have applications in various situations where only partial information -either kinetic or geometric -on trajectories is accessible ; in addition, it appears to be a useful technical tool that for instance can give access to persistence properties of self interacting random walks.

FIG. 1 :

 1 FIG.1:(a) Starting from x0, the random walker crosses 0 for the first time at time t, having explored up to a distance x from the origin. The joint law σ(x, t|x0) is the density probability function of such joint events. (b) Consider a random searcher evolving in a (greyed out) domain, filled with Poisson distributed targets. Having as only information the time t of exit from the domain, we display the probability Pt of encounter with at least one target in terms of the rescaled variable ρ dw t in the case of a Brownian searcher of diffusion coefficient D = 1/2. Numerical integration of the exact result (13) (symbols) and asymptotic scaling form (15) (dashed line) are shown.

FIG. 2 :

 2 FIG. 2: Conditional distribution f ) of the rescaled variable τ (see text). Distributions are drawn for fixed s (discrete space) or x (continuous space) and collapse. A,B -Markovian Processes ; C,D Non Markovian Processes. See SM for details on simulations.

  1-β β -3 f SAT W (τ ) where h(1) = Γ(-2+2/β) . Finally, the conditional distribution f SAT W is defined by its strikingly simple Laplace transform:fSAT W (p) = ∞ 0 e -pu f SAT W (u)du =