Dr Rémi F André

Dr Alberto Palazzolo

Dr Cyprien Poucin

Dr François Ribot

Dr Sophie Carenco
email: sophie.carenco@sorbonne-universite.fr

Phosphine-Catalyzed Activation of Phenylsilane for Benzaldehyde Reduction

Keywords: Hydrosilylation, hypervalent silicon, Lewis base, organocatalysis, phosphine 10. References

published or not. The documents may come

Introduction

As a synthetic strategy, hydrosilylation reactions are widely used in industry for the transformation of alkenes to organosilanes, [1] and more generally explored in academia for the hydrogenation of carbonyl compounds to primary alcohols, [2] and that of CO 2 to formic acid or methanol. [3] Along with hydroboranes, hydrosilane reagents, R n SiH 4-n (R = alkyl, aryl, alkoxy, halo), attracted interest as most of them are chemically stable, readily available and easy to handle: they serve as hydride source via the polarized Si (δ+) -H (δ-) bond. [START_REF] Rendler | Mod. Reduct. Methods[END_REF] In the context of carbonyl reduction, the addition of Si-H to the C=O bond leads to the formation of a silyl ether in a single step under mild conditions, which can be further hydrolyzed to form an alcohol. Applications in industrial settings generally favor the use of cheap PMHS (polymethylhydrosiloxane, (-SiMeHO-) n ), or TMDS (tetramethyldisiloxane, HMe 2 SiOSiMe 2 H), but academic works often report reactions with Et 3 SiH, (EtO) 3 SiH, Ph 2 SiH 2 , PhMe 2 SiH or PhSiH 3 , more reactive for "proof of concept" studies and less prone to form siloxane gels. [START_REF] Addis | [END_REF] Among possible paths, the activation of the silane can rely on transition metal complexes or nanoparticles via an oxidative addition resulting in two moieties, R 3 Si - and H -, coordinated to the metal center, such as platinum-based Speier's and Karstedt's catalysts that are the industrial standards. [START_REF] Rendler | Mod. Reduct. Methods[END_REF]6] A large library of transition metal complexes has been explored for the hydrosilylation of carbonyls, including noble metals but also first-row transition metals such as manganese, iron, cobalt and nickel. [7] The high activity of some of them is evidenced by the mild conditions used, as well as the low catalytic loadings, down to 0.02 mol %. Using transition metals obviously comes with advantages in terms of chemo-and stereoselectivities via the fine tuning of the catalyst structure, but is also limited by the cost of the metals, the synthesis of the ligands or the removal of the metal traces for the following steps. [8] Alternative organocatalytic pathways were therefore developed, recently reviewed by Nikonov et al., [8] and Hreczycho et al. [2] The catalysis mechanism may consist in the electrophilic activation of the aldehyde with Brønsted or Lewis acids, via the weakening of the C=O bond, in the electrophilic activation of the silane with Lewis acids, via the weakening of the Si-H bond, or in the nucleophilic activation of the silane with Lewis bases, via the formation of a hypervalent silicate [R 3 (base)SiH] (-) complex. [9] Works with Lewis bases were mostly performed with anions such as fluorides, [10][11][12] alkoxides, [9,13,14] hydroxides, [14] or carbonates [15,16] (Figure 1), but more rarely with neutral molecules, yet more adapted to organic solvents. [17,18] Another possible Lewis base substitute consists in phosphines, which have attracted interest these past decades as standalone organocatalysts. [19] While these suffer from limitations regarding their practical use, such as air-sensitivity, the possibility to explore their reactivity as a family of molecules rather than as stand-alone catalysts, e.g. by gradually varying critical parameters like steric hindrance and nucleophilicity, seems appealing for mechanistic understanding of phosphine reactivity in general.

Figure 1.

Selection of reported Lewis base-catalyzed hydrosilylations of unsaturated bonds (aldehydes and nitriles) from references [11] , [12] , [14] , [9] , [16] and [15] (from top to bottom).

In this work, we report the metal-free hydrosilylation of benzaldehyde with PhSiH 3 using tertiary phosphines as Lewis bases. The optimization of the reaction conditions is first presented, in particular as to the solvent and the nature of phosphine. The products were partially identified by means of 1 H- 29 Si NMR experiments, and their formation was followed

Results and discussion

Hydrosilylation reaction. Blank reactions were first performed. The catalyst-free (no phosphine) hydrosilylation of benzaldehyde (PhCHO, 2.5 equiv.) with phenylsilane (PhSiH 3 , 1 equiv.) was performed in dry acetonitrile (ACN) and did not lead to conversion of either the silane or the aldehyde. Besides, benzaldehyde and trimethylphosphine (PMe 3 ) did not react together in ACN (Figure S3). However, a solution of PhSiH 3 and PMe 3 turned slightly yellow in a few minutes and bubbling was observed, indicating that a reaction was taking place, leading to production of H 2 as indicated by 1 H NMR (singlet signal at 4.57 ppm) (Figure S4). [20] Dihydrogen is thought to originate from the phosphine-catalyzed decomposition of PhSiH 3 with traces of protic molecules (e.g. water). [21] The catalytic hydrosilylation was then studied with P n Bu 3 as the catalyst. In dry ACN, the reaction of PhCHO (0.2 mol/L) with PhSiH 3 (0.08 mol/L), introduced in slight excess (0.4 equiv. versus PhCHO), proceeded smoothly in the presence of P n Bu 3 (10 mol % versus the silane), to afford silylethers overnight at room temperature (Figure 2A). The progression of the reaction was characterized by 1 H NMR with the decrease of the aldehydic peak of PhCHO and the growth of "benzyloxy" (Ph-CH 2 -O) singlets in the 4.65-4.25 ppm region (Figure S5A). Approximately half of the benzaldehyde was converted in 16 h, with an error bar estimated to be ca. 10 % (conversions ranging from 35 % and 52 % for four repetitions of the experiment) ( [22] no gas evolution nor modification of the NMR spectrum were observed in the present case, ruling out this possibility. The benzaldehyde hydrosilylation did not proceed in this solvent neither (Table 1, entry 6). Benzonitrile was used as a second nitrile solvent and led to a limited conversion of 7 % (Table 1, entry 7). Anhydrous propylene carbonate (PC), a highly polar and aprotic solvent (ε r = 65), led to a quasi-quantitative conversion (Table 1, entry 8). Mixtures of ACN and toluene were finally investigated: the higher the volume fraction of ACN, the higher the conversion (Table 1, entries 9-10).

As similar products were obtained for ACN and propylene carbonate, we concluded that the chemical reactivity of the C≡N bond was not involved. Rather, the impact of the solvent was limited to its physical properties (e.g. dielectric constant, polarity). Although propylene carbonate appeared to be an efficient solvent, ACN was used in the following to facilitate the interpretation of the 1 H NMR data. Interestingly, the relative polarity according to Miller's chart was not a suitable parameter to rationalize the evolution of the conversion (Table 1 and Figure S1): conversions from 0 % to 97 % were found for close relative polarities, from 0.40 to 0.55. Interestingly, the conversion was found to increase with the relative dielectric constant ε r (Table 1 and Figure 2B). This was consistent with charged intermediates and/or transition states being formed during the reaction, which will be discussed later.

Entry Solvent

Relative polarity (Miller's chart) [a] Relative dielectric constant (ε r )

Benzaldehyde conversion 1. Solvent screening for benzaldehyde hydrosilylation. Reaction conditions: PhSiH 3 (50 µL, 0.4 mmol, 1 equiv.), PhCHO (100 µL, 1 mmol, 2.5 equiv.), mesitylene (internal standard) (55 µL, 0.4 mmol, 1 equiv.), P n Bu 3 (10 µL, 0.04 mmol, 0.1 equiv.), solvent (5 mL), r.t., 24 h. [a] Relative polarity from reference. [23] [b] The benzaldehyde conversion was set as the ratio of half of the "benzyloxy" protons versus the benzaldehyde leftover + half of the "benzyloxy" protons, determined by 1 H NMR. [c] The "benzyloxy protons" in 1 H NMR are masked by the signal of the solvent, the benzaldehyde conversion was determined via a 1 H NMR measurement before and after P n Bu 3 addition. [d] Polarities were estimated as linear combinations of those of toluene and ACN, and dielectric constants extracted from reference. [24] Figure 2. (A) Hydrosilylation of benzaldehyde catalyzed by P n Bu 3 (reaction conditions as outlined in Table 1). (B) Dependance over the polarity of the solvent (relative dielectric constant ε r ). Open squares correspond to toluene, ACN and mixtures thereof (volume ratio indicated). THF: tetrahydrofuran, DMC: dimethylcarbonate, ACN: acetonitrile.

Phosphine screening. The impact of the chemical nature of the phosphine was then evaluated with a range of alkyl-and arylphosphines, as well as with a phosphine oxide and a phosphite (Table 2 and Figure 3A). All the reactions were performed in ACN with a catalyst loading of 10 mol %. A moderate to high conversion of benzaldehyde was obtained with three linear trialkylphosphines, PMe (50 µL, 0.4 mmol, 1 equiv.), PhCHO (100 µL, 1 mmol, 2.5 equiv.), mesitylene (internal standard) (55 µL, 0.4 mmol, 1 equiv.), catalyst (0.04 mmol, 0.1 equiv.), acetonitrile (5 mL), r.t., 24 h. [a] The benzaldehyde conversion was set as the ratio of half of the "benzyloxy" protons versus the benzaldehyde leftover + half of the "benzyloxy" protons, determined by 1 H NMR. [b] Average result on two runs. [c] Average result on four runs.

The nature of the substituents has a dramatic impact on the basicity (Lewis and Brønsted) of the phosphorus atom because of a combination of steric hindrance and of electronic effect. Classifying the phosphines as to their catalytic activity is generally difficult when using a sole descriptor such as the Tolman cone angle (characteristic of the steric hindrance), the Tolman Electronic Parameter (TEP) (characteristic of the electron donation from the phosphorus atom), the energy level of the HOMO or the pKa of the Brønsted acid/base couple. Accordingly, the benzaldehyde conversion plotted as a function of the Tolman cone angles, as historically reported (Figure S2A), [25] or recomputed by Jover et al.

(Figure S2B), [26] or as a function of the TEP (Figure S2C), [25] does not follow any clear trend. As the recomputation of the Tolman cone angles by Jover et al. in 2019 sensibly updated the values for a few phosphines such as P n Bu 3 (from 132° to 160°, Table S1), the He8_steric parameter was finally chosen to classify the steric hindrance around the phosphorus atom. The values correspond to the destabilization energy (in kcal/mol) of the interaction of the ligand with a ring of eight helium atoms and give rise to a steric hindrance scale in arbitrary units. [26][27][28] The corresponding plot is on Figure S2D and here also, no clear trend was observed.

At this stage, it was concluded that a simple trend with electronic or steric effects was not able to capture the complexity of the situation. To discuss crossed impact of steric and electronic effects, Tolman proposed to plot the studied parameter (in our case, the conversion) on stereo-electronic maps, i.e., with the Tolman cone angle and the TEP as variables. [25] In the present work, the He8_steric parameter was used to describe the steric hindrance, instead of the Tolman cone angle (Figure 3B). The highest conversions were obtained for phosphines presenting both low steric hindrance and low TEP, i.e., with a pronounced electron-donor character from the phosphorus center (bottom left region of Figure 3B), which would coincide with a strong nucleophilic character. The series {PMe 3 , PMe 2 Ph, PMePh 2 , PPh 3 } with the gradual substitution of the methyl groups by phenyl ones, leading to a total loss of catalytic activity for PMePh 2 and PPh 3 , clearly illustrates that alkylphosphines are better catalysts than arylphosphines for this reaction (zone I in Figure 3B). The C sp 3 orbital at the origin of the P-C bond in the alkylphosphines has indeed a higher energy than the equivalent C sp 2 orbital in the arylphosphines. Upon interaction with the orbitals of phosphorus, the resulting filled orbital of the P-C bond is therefore more centered on the phosphorus atom in alkylphosphines, making them better electron donors than arylphosphines. Accordingly, the Brønsted basicity of the phosphine decreases in the order {PMe 3 , PMe 2 Ph, PMePh 2 , PPh 3 } with pKa of respectively 8.6, 6.5, n.d. and 2.7, [29,30] as well as their nucleophilicity, ranked by Buckler et al., from 2.24 (a.u.) for PMe 3 to 0.04 for PPh 3 . [31] In this series, the trends for both the steric hindrance and the electron donation from the phosphorus atom go along: a strongly donating and poorly hindered phosphine is favorable.

Additionally, the series {P n Bu 3 , P i Bu 3 , P t Bu 3 } with tributylphosphines of diverse branching demonstrates a negative impact of bulky groups on the benzaldehyde conversion (zone II in Figure 3B). The nature of the alkyl group has little impact on the Brønsted basicity of the trialkylphosphines, which is of the same order of magnitude for P n Bu 3 , P i Bu 3 , P t Bu 3 and PMe 3 , [29,30,32] whereas the nucleophilicity by Buckler et al. decreased from 1.62 (a.u.) for P n Bu 3 to 0.14 for P i Bu 3 . [31] This trend matches with our experimental observations: the apparent critical factor for the catalytic activity is the nucleophilicity, well anticipated from the Lewis basicity. Besides, P t Bu 3 seems to be too sterically hindered for the reaction to proceed.

The order of reactivity unveiled in the present work (P(OPh) 3 , PPh 3 << PCy 3 < P n Bu 3 ) is coherent with Mayr's attempt at ranking the nucleophilicity of phosphines via the use of four parameters: the σ-donor capacity (χ d ), the cone angle (θ), the secondary electronic effect (E ar ) and the π-electron acceptor capacity (π p ). Interestingly, the comparison with other types of nucleophiles suggests that butylamine and piperidine would be stronger Lewis bases than P n Bu 3 and thereby could be used for the activation of PhSiH 3 . [33] Overall, the phosphine screening revealed that a low steric hindrance and a high nucleophilicity are required for the reaction to proceed with phenylsilane and benzaldehyde.

Product identification. Phenylsilane may provide up to three hydrides and thereby hydrogenate three benzaldehyde molecules. This sequential reactivity of PhSiH 3 results in three products PhSiH 3-n (OBn) n (n = 1, 2, 3) whose identification would give insight into the mechanisms via the time monitoring of the different species. The corresponding 1 H and 29 Si chemical shifts were determined via a combination of 1 H and 1 H- 29 Si sequences (Figure 4 and Table 3, see Supplementary Information for details). To the best of our knowledge, the 29 Si chemical shifts were not referenced for the synthesized benzyloxysilanes, but that of PhSi(OBn) 3 (-56.7 ppm) is coherent with the shift reported for a similar compound, PhSi(OEt) 3 (-55.6 ppm). [34] The evolution of the 29 Si chemical shift in the series is delicate to rationalize as it corresponds to the substitution of a hydride "ligand" H -by an alkoxy one S2). Among the three major ones, US1 and US2 present at least one Si-H bond, to the contrary of the third one, US3 (Figure S9). Besides, it appeared that US1 and US2 further reacted as the intensity of their peaks decreased after a few hours, while that of US3 only increased (Figure S10-11). We therefore suggest that these species also follow a reaction pattern ruled by successive hydrosilylations of benzaldehyde molecules. Neither the nature of these molecules nor the conditions of their production could be formally identified.

We nonetheless suggest they may be linked to a ligand rearrangement around the silicon atom, as reported for alkoxy-, alkyl-and arylsilanes in the presence of nucleophiles, [9,35] or to a partial hydrolysis due to water traces. and phenylsilanetriol PhSi(OH) 3 , which further condensed as polymers, hence the absence of signal in 1 H NMR. [36] The amount of produced H 2 indeed corresponds to slightly less than three times that of consumed PhSiH 3 . The decrease of the benzaldehyde concentration, while no production of benzyl alcohol was recorded, suggests either a further reaction with the soformed polymer, or a partial hydrosilylation of PhSiH 3 in PhSiH 2 (OBn), directly followed by the hydrolysis of the remaining hydrides and condensation of the formed silanol. The identification of these hypothetic polymeric structures is nonetheless out of the scope of the present work. The end of the production of H 2 marked the appearance of PhSiH 2 (OBn) and a faster consumption of PhSiH 3 and PhCHO, we therefore hypothesize that the activated silanes by the phosphine were available to react with benzaldehyde only once all the labile protons had reacted.

Time evolution of

For an initial concentration in PhSiH 3 of 80 mmol/L, the concentration in PhSiH 2 (OBn) increased rapidly up to 11 mmol/L after 4 h of reaction and then decreased down to 2 mmol/L in the following hours (Figure 5A). The production of PhSiH(OBn) 2 started immediately with that of PhSiH 2 (OBn) and its concentration reached 43 mmol/L after 18 h, before slowly decreasing to the benefit of PhSi(OBn) 3 . It is noteworthy that the sum of the estimated concentrations of the four species PhSiH 3-n (OBn) n (dotted line on Figure 5A)

was roughly constant after a drop by 10 % during the induction time. At the maximal concentration recorded for PhSiH 2 (OBn) after 4 h, the production of PhSiH 2 (OBn) from PhSiH roughly compensated its consumption for a further hydrosilylation to form PhSiH(OBn) 2 , hence a plateau in concentration was observed. If we hypothesize the hydrosilylation mechanism is similar for the three successive reactions, the expression of the reaction rate displays the same orders of reaction respectively to the silane, the benzaldehyde and the phosphine (Figure S20). At all times, the part of the reaction rate linked to the concentration in benzaldehyde and in phosphine is identical for the first and the second steps.

Therefore, the maximum concentration of PhSiH S3). As expected, the reaction times at which these plateaus were observed were smaller when the PMe 3 concentration was larger, indicating a non-null partial order with respect to the phosphine.

The superior reactivity of PhSiH 2 (OBn) compared with PhSiH 3 and PhSiH(OBn) 2 is in line with literature work using a cobalt complex for the three successive hydrosilylation reactions, though no justification was proposed. [37] The benzaldehyde and benzyloxy groups concentrations were estimated on the basis of the integration in 1 H NMR of respectively the aldehydic peak at 10 ppm (Figure S5A) and the benzyloxy region, i.e., 5-4.5 ppm in CD 3 CN (Figure 4). The decrease in PhCHO concentration was clearly correlated with the increase of that of the benzyloxy groups (red vs.

solid green in Figure 5B). Roughly 80 % of the benzyloxy peaks may be accounted to the different products of the hydrosilylation PhSi 3-n (OBn) n (dotted green in Figure 5B), indicating the presence of secondary or degradation products. This percentage was however dependent on the reaction and was of 60 % and 85 % for respectively the 17 mol % and 80 mol % phosphine loadings (Figures S14B andS17B). The workup of these secondary products bearing benzyloxo groups is nonetheless expected to eventually lead to the desired alcohol as well.

Mechanistic considerations. The strong dependence of the benzaldehyde conversion over the dielectric constant of the solvent suggests that charges are developed on the activated complex of the rate-limiting step of the reaction. Besides, a screening over different phosphines established that the best catalysts are Lewis bases that present a limited steric hindrance and a high nucleophilicity at the phosphorus center. It should be noted that if a phosphine catalyzes the reaction, the corresponding phosphine oxide does not. Indeed, when PMe 3 was oxidized by exposure to air, no further conversion was detected.

All these observations are coherent with a mechanism, showed in Figure 6, during which the phosphine PR 3 interacts with the silane 1 with the lone pair of the phosphorus, in order to form a penta-valent silicate 2, with partial charges δ+ and δdeveloped on respectively the phosphorus and the silicon atom. A reorganization of the electronic density occurs in penta-valent silicon species leading to a lengthening of the bonds between Si and the peripheral ligands (H -here) and an enhancement of the nucleophile character of these last ones. [START_REF] Rendler | Mod. Reduct. Methods[END_REF]38] The first step is therefore followed by an attack of the activated hydride H -on the An alternative mechanism may consist in the coordination of the benzaldehyde to the silicon center of 2 via the oxygen atom to form a hexa-valent silicon species, followed by an intramolecular attack of a hydride. While the attack of the Lewis basic phosphine on the silicon center seems natural, the Lewis acidity of the negatively charged silicon center in 2 is unclear. Such a behavior would be described as the Lewis base activation of silicon Lewis acids, and was suggested in particular by Oestreich and Denmark for the activation of silanes with fluoride and carbonate ions. [START_REF] Rendler | Mod. Reduct. Methods[END_REF]38]

Conclusion

In summary, we reported herein the activation of phenylsilane catalyzed by simple neutral organic bases such as PMe

Table of Content

Trialkylphosphines were used as organocatalysts for the hydrosilylation of benzaldehyde to arylbenzyloxyhydrosilanes at room temperature. The products were obtained in yields up to 97 %. The nucleophilicity of the catalyst was demonstrated to be critical via a stereoelectronic map and a Lewis base activation mechanism with a hypervalent silicon intermediate species was proposed.

Twitter handle: @SophieCARENCO

using 1 H

 1 NMR, giving insight into the reaction mechanism and the relative reactivity of the reagents. While trialkylphosphines are not expected to substitute classical catalysts such as K 2 CO 3 or TBAF (tetra-n-butylammonium fluoride), this works offers a new entry point to the design of alternative organocatalysts in a variety of solvents and provides tools to rationalize the mechanisms at stake in hydrosilane Lewis-base activation. Besides, such a catalytic activity displayed by a phosphine alone may shed a new light on similar works reporting the use of transition metal complexes coordinated by phosphine ligands.

Figure 3 .

 3 Figure 3. (A) Phosphine screening for the hydrosilylation of benzaldehyde catalyzed in ACN (reaction conditions as outlined in Table2) and corresponding (B) stereo-electronic map. The color code indicates the benzaldehyde conversion for several phosphines (green: > 40 %, orange: < 15 %, red: 0 %). Two zones (labeled I and II) are plotted as grey ellipses as a guide for the eye.

  BnO -, both transferring electron density toward the silicon atom. Accordingly, the values measured for the two intermediate compounds, PhSiH 2 (OBn) (-66.6 ppm) and PhSiH(OBn) 2 (-28.2 ppm) are outside the range defined by PhSiH 3 and PhSi(OBn) 3 (respectively -59.8 ppm and -56.7 ppm). The possibility of an aliasing in the 1 H-29 Si experiments that would lead to an incorrect29 Si position was ruled out with an experiment with a larger spectral width (SW) in the indirect dimension.

Figure 4 .

 4 Figure 4. Details of the 1 H NMR spectrum of a solution of PhCHO, PhSiH 3 and PMe 3 in acetonitrile after 2 h, with assignment of the species PhSiH 3-n (OBn) n (n = 1, 2, 3).

  the reaction products. The hydrosilylation reaction was monitored by1 H NMR in deuterated acetonitrile at different catalyst loadings (10 mol %, 17 mol % and 80 mol % of PMe 3 ) for mechanistic purposes. The evolution of the three compounds of general formula PhSiH 3-n (OBn) n (n = 1, 2, 3) was similar in the three experiments and will only be commented in the case of 10 mol % (Figure5) (see FigureS14-19 for 17 mol % and 80 mol %). First, an induction period of 2 h (shorter for catalyst loadings of 17 mol % and 80 mol %) was observed during which the concentrations in PhSiH 3 and PhCHO decreased by respectively ca. 15 mmol/L and ca. 20 mmol/L, Meanwhile, the concentration of H 2 increased by ca. 40 mmol/L, while merely no other compound was detected by1 H NMR (FigureS12, red for H 2 and blue for PhSiH 3-n (OBn) n (n = 1, 2, 3)). The phosphine possibly activated the silane molecules to react with adventitious water to form H 2

Figure 5 .

 5 Figure 5. Time monitoring of benzaldehyde hydrosilylation catalyzed by PMe 3 (10 mol %). (A) Concentrations of the phenylsilane and of the three hydrosilylation products, and sum of the four species (dotted line). (B) Concentrations in benzaldehyde, integration of the benzyloxy region and sum of the benzyloxy groups from PhSiH 3-n (OBn) n for n = 1, 2, 3 (dotted line).

aldehydic carbon of benzaldehyde to form the ion pair 3

 3 which further reacts to form 4. A concerted mechanism of the attack of the hydride and of the formed alkoxide on the silicon center to yield the compound 4 directly from 2 cannot be ruled out, especially considering the high reactivity of the hypothetically formed silylium cation. The intermolecular hydride transfer is expected to be the limiting step, in agreement with the crucial role of the polarity of the solvent. The dissociation of PMe 3 finally leads to the benzyloxysilane 5. The higher reactivity observed for PhSiH 2 (OBn) compared to PhSiH 3 and PhSiH(OBn) 2 may be due to a competition of two antagonist effects: PhSiH 2 (OBn) is more Lewis acidic than PhSiH 3 but less sterically hindered than PhSiH(OBn) 2 . This interplay of sterics and electronic allows for more efficient phosphine coordination to PhSiH 2 (OBn).

Figure 6 .

 6 Figure 6. Proposed mechanism for the hydrosilylation of benzaldehyde catalyzed by trimethylphosphine (PMe 3 ).

  3 and P n Bu 3 . The reaction consists in the hydrosilylation of benzaldehyde with PhSiH 3 , performed at room temperature. The polarity of the solvent is beneficial to the reaction and the highest yields were obtained in acetonitrile (ε r = 37) and propylene carbonate (ε r = 65), with conversions of 46 % and 97 %, respectively. A screening over 13 phosphines indicated that the nucleophilicity of the phosphine allows the silane activation, with an optimum in the case of PMe 3 . The successive stages of the hydrosilylation of benzaldehyde with phenylsilane, PhSiH 3-n (OBn) n (n = 1, 2, 3), were identified through a combination of 1 H-29 Si experiments and this attribution allowed a time monitoring of the concentration of the species. The most notable point is the higher reactivity of PhSiH 2 (OBn) compared to PhSiH 3 and PhSiH(OBn) 2 . The dependance over the solvent and the phosphine is consistent with a mechanism of Lewis base activation of the silicon Lewis acid. The conditions presented herein, i.e., the use of PhSiH 3 and of air-sensitive trialkylphosphines, are not designed for large scale applications but they open a way to the use of neutral organic bases, which may bring interest to the catalysis community for further development of alternative catalysts (e.g. amines) and for mechanistic considerations.

Table

  1, entry 1). The identification of the multiple products detected after reaction will be discussed later, taking into account that the reaction intermediates are expected to exhibit different reaction rates. Overall, this first set of experiments confirmed that P n Bu 3 was a suitable catalyst.

	Solvent screening. Varying the solvent is an interesting manner to obtain insights into
	the reaction mechanism. In particular, the charges developed by transition states or reaction
	intermediates were expected to be better stabilized in polar solvents, leading to higher
	conversions. A screening over solvents of various polarities and protic characters was
	performed (Table 1). No benzaldehyde conversion was recorded for reactions run in the less
	polar solvents: n-hexane, toluene, dimethylcarbonate (DMC), and tetrahydrofuran (THF)
	(Table 1, entries 2-5). Although isopropanol (iPrOH) was reported to react with PhSiH 3 in the
	presence of a suitable catalyst to form H 2 and Ph(iPrO)SiH 2 ,

Table

					[b]
	1	ACN	0.46	37	46 %
	2	n-hexane	0.01	1.9	0 %
	3	Toluene	0.10	2.4	0 %
	4	DMC	0.23	3.1	0 %
	5	THF	0.20	7.6	0 %
	6	iPrOH	0.55	20	0 %
	7	Benzonitrile	0.33	26	7 %
	8 [c]	Propylene carbonate	0.47	65	97 %
	9 [d]	ACN/Toluene (5:5)	0.28	25	2 %
	10 [d]	ACN/Toluene (9:1)	0.42	33	25 %

Table 2 ,

 2 3 (88 %), P n Bu 3 (46 %) and P n Oct 3 (56 %), and one alkylarylphosphine, PMe 2 Ph (12 %) (Table2, entries 1, 2, 5 and 8). Low conversions were recorded with three other trialkylphosphines (P i Bu 3 , P t Bu 3 and PCy 3 ) and two secondary entries 3, 4 and 12-14).

	Entry	Catalyst (10 mol %)	Benzaldehyde conversion [a]
	1 [b]	PMe 3	88 %
	2 [b]	PMe 2 Ph	12 %
	3	PMePh 2	0 %
	4	PPh 3	0 %
	5 [c]	P n Bu 3	46 %
	6 [b]	P i Bu 3	6 %
	7	P t Bu 3	1 %
	8 [b]	POct 3	56 %
	9 [b]	PCy 3	5 %
	10	PHPh 2	2 %
	11	PH t Bu 2	6 %
	12	dppb	0 %
	13	P(OPh) 3	0 %
	14	PHPh 2 O	0 %

phosphines, PH t Bu 2 and PHPh 2 (Table

2

, entries 6, 7 and 9-11), while no conversion was obtained with di-and triarylphoshines (PMePh 2 , PPh 3 , 1,4-bis(diphenylphosphino)butane (dppb)), triphenylphosphite (P(OPh) 3 ) and diphenyl phosphine oxide (PHPh 2 O) (

Table 2 .

 2 Catalyst screening for benzaldehyde hydrosilylation. Reaction conditions: PhSiH 3

Table 3 .

 3 1 H and29 Si chemical shifts of phenylsilane and of the three formed benzyloxysilanes PhSiH 3-n (OBn) n . The CH 3 CN peak was calibrated at 1.96 ppm.

	Species	29 Si (ppm)	Si -1 H (ppm)	-C 1 H 2 -(ppm)
	PhSiH 3	-59.8	4.18	n.a.
	PhSiH 2 (OBn)	-66.6	5.05	4.82
	PhSiH(OBn) 2	-28.2	5.07	4.89
	PhSi(OBn) 3	-56.7	n.a.	4.90
	Depending on the reaction conditions, unidentified species (US) were detected in
	limited amounts with characteristic

1 H signals ca. 6 ppm coupling with carbons at ca. 98 ppm (Figure S5-6 and Table

  2 (OBn) only depended on the concentrations of the silanes PhSiH 3 and PhSiH 2 (OBn) and of the corresponding reaction rate constants k 0 and k 1 , respectively. As this maximum was reached at a lower concentration in PhSiH 2 (OBn) (11 mmol/L) than in PhSiH 3 (51 mmol/L), we can conclude that PhSiH 2 (OBn) is more reactive than PhSiH 3 and that k 0 is lower than k 1 . A ratio of 0.22 was obtained for the relative reaction rates values k 0 /k 1 . A similar analysis on the plateau of PhSiH(OBn) 2 after 18 h ([PhSiH 2 (OBn)] = 2 mmol/L, [PhSiH(OBn) 2 ] = 43 mmol/L), led us to the conclusion that k 1 is higher than k 2 , with a k 1 /k 2 ratio of 22. Results of similar orders of magnitude were observed for the reactions run with 17 mol % of PMe 3 and 80 mol % of PMe 3 with ratios k 0 /k 1 of 0.31 and 0.10, respectively, and ratios k 1 /k 2 of 4.6 and 16, respectively (Table

Acknowledgements

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.

758480). This work was supported by Sorbonne

Experimental section

Catalytic tests. In a glovebox under argon (< 0.5 O 2 ppm, < 0.5 H 2 O ppm), a 10 mL vial was loaded with dry acetonitrile (5 mL), phenylsilane (50 µL, 0.4 mmol, 1 equiv.), benzaldehyde (102 µL, 1 mmol, 2.5 equiv.), mesitylene (internal standard) (55 µL, 0.4 mmol, 1 equiv.) and PR 3 (0.04 mmol, 0.1 equiv.). The reagents were taken with a glass micro syringe, with an error of ± 10 %. The reaction was stirred at room temperature for 24 h. Gas formation was observed for the more concentrated reactions and in propylene carbonate. An aliquot of the liquid phase was collected for analysis by 1 H NMR in CDCl 3 . All NMR kinetic experiments were performed under argon atmosphere using J. Young NMR tubes equipped with Teflon valves directly on the reaction crude with the use of CD 3 CN instead of CH 3 CN.

Multinuclear NMR spectroscopy. All NMR experiments were performed on a

Bruker Avance-III 300 spectrometer (300.13, 75.47 and 59.63 MHz for 1 H, 13 C and 29 Si, respectively) equipped with a BBFO 5mm grad z probe. 29 Si-{ 1 H} NMR experiments were carried out with an inverse gated decoupling to avoid NOE effect that can lead to a loss of signal because of the negative gyromagnetic ratio of 29 Si. Gradient enhanced HSQC ( 1 H- 13 C and 1 H-29 Si) and HMBC ( 1 H- 29 Si) experiments were performed with sequences directly taken from the Bruker's library (hsqcetgp and hmbcetgpnd). For 1 H- 13 C HSQC, a J coupling value of 145 Hz was chosen. For 1 H- 29 Si HSQC and 1 H- 29

Conflict of interest

The authors declare no conflict of interest.


Florent Benaych-Georges

ON A SURPRISING RELATION BETWEEN THE MARCHENKO-PASTUR LAW, RECTANGULAR AND SQUARE FREE CONVOLUTIONS

Keywords: . 2000 Mathematics Subject Classification. 46L54, 15A52 Free Probability, Random Matrices, Free Convolution, Infinitely Divisible Laws, Marchenko-Pastur Law

In this paper, we prove a result linking the square and the rectangular R-transforms, the consequence of which is a surprising relation between the square and rectangular versions the free additive convolutions, involving the Marchenko-Pastur law. Consequences on random matrices, on infinite divisibility and on the arithmetics of the square versions of the free additive and multiplicative convolutions are given.

Dans cet article, on prouve un résultat reliant les versions carré et rectangulaire de la Rtransformée, qui a pour conséquence une relation surprenante entre les versions carré et rectangulaire de la convolution libre additive, impliquant la loi de Marchenko-Pastur. On donne des conséquences de ce résultat portant sur les matrices aléatoires, sur l'infinie divisibilité et sur l'arithmétique des versions carré des convolutions additives et multiplicatives.

Free convolutions are operations on probability measures on the real line which allow to compute the empirical spectral 1 or singular 2 measures of large random matrices which are expressed as sums or products of independent random matrices, the spectral measures of which are known. More specifically, the operations ⊞, ⊠, called respectively free additive and multiplicative convolutions are defined in the following way [START_REF] Voiculescu | Free random variables CRM Monograghs Series[END_REF]. Let, for each n, M n , N n be n by n independent random hermitian matrices, one of them having a distribution which is invariant under the action of the unitary group by conjugation, the empirical spectral measures of which converge, as n tends to infinity, to non random probability measures denoted respectively by τ 1 , τ 2 . Then τ 1 ⊞ τ 2 is the limit of the empirical spectral law of M n + N n and, in the case where the matrices are positive, τ 1 ⊠ τ 2 is the limit of the empirical spectral law of M n N n . In the same way, for any λ ∈ [0, 1], the rectangular free convolution ⊞ λ is defined, in [B09], in the following way. Let M n,p , N n,p be n by p independent random matrices, one of them having a distribution which is invariant by multiplication by any unitary matrix on any side, the symmetrized 3 empirical singular measures of which tend, as n, p tend to infinity in such a way that n/p tends to λ, to non random probability measures ν 1 , ν 2 . Then the symmetrized empirical singular law of M n,p + N n,p tends to ν 1 ⊞ λ ν 2 . These operations can also, equivalently, be defined in reference to free elements of a non commutative probability space, but in this paper, we have chosen to use the random matrix point of view.

In the cases λ = 0 or λ = 1, i.e. where the rectangular random matrices considered in the previous definition are either "almost flat" or "almost square", the rectangular free convolution with ratio λ can be expressed with the additive free convolution: ⊞ 1 = ⊞ and for all symmetric laws ν 1 , ν 2 , ν 1 ⊞ 0 ν 2 is the symmetric law the push-forward by the map t → t 2 of which is the free additive convolution of the push forwards of ν 1 and ν 2 by the same map. These surprising relations have no simple explanations, but they allow to hope a general relation between the operations ⊞ λ and ⊞, which would be true for any λ. Up to now, despite many efforts, no such relation had been found, until a paper of Debbah and Ryan [START_REF] Debbah | Multiplicative free Convolution and Information-Plus-Noise Type Matrices[END_REF], where a relation between ⊞ λ , ⊞ and ⊠ is proved in a particular case. In the present paper, we give a shorter proof of a wide generalization 4 of their result: for any λ ∈ (0, 1], we define µ λ to be the law of λ times a random variable with law the Marchenko-Pastur law with mean 1/λ, and we prove that for any pair µ, µ ′ of probability measures on [0, +∞), we have

(1) µ ⊠ µ λ ⊞ λ µ ′ ⊠ µ λ = (µ ⊞ µ ′ ) ⊠ µ λ ,
where for any probability measure ρ on [0, +∞), √ ρ denotes the symmetrization of the pushforward of ρ by the map t → √ t. Our proof is based on the following relation between the Rtransform 5 R µ of a probability measure µ on [0, +∞) and the rectangular R-transform C √ µ⊠µ λ 1 The empirical spectral measure of a matrix is the uniform law on its eigenvalues with multiplicity. 2 The empirical singular measure of a matrix M with size n by p (n ≤ p) is the empirical spectral measure of

|M | := √ M M * . 3
The symmetrization of a law µ on [0, +∞) is the law ν defined by ν(A) = µ(A)+µ (-A) with ratio λ of √ µ ⊠ µ λ : we prove that for all z,

R µ (z) = C √ µ⊠µ λ (z).
This relation also allows us to prove precise relations between ⊞-infinitely divisible laws and ⊞ λ -infinitely divisible laws.

We would like to observe that formula (1) has some consequences which are far from obvious. It means that for n, p large integers such that n/p ≃ λ, for A, B, M, M ′ independent random matrices with respective sizes n × n, n × n, n × p and n × p such that A, B are invariant in law under left and right multiplication by unitary matrices and M, M ′ have independent Gaussian entries, then as far as the spectral measure is concerned,

(AM + BM ′ )(AM + BM ′ ) * ≃ AM (AM ) * + BM ′ (BM ′ ) * .
It also means, if 1 << n << p, that for X, Y independent n × p random matrices, as far as the spectrums are concerned,

(X + Y )(X + Y ) * ≃ XX * + Y Y * .
The relation (1) has also consequences on the arithmetics of free additive and multiplicative convolutions ⊞ and ⊠ (Corollaries 7 and 12) which wasn't known yet, despite the many papers written the last years about questions related to this subject, e.g. [BV95, BPB99, B04, CG08a, CG08b, BBG08, BBCC08].

Acknowledgments:

The author would like to thank his friend Raj Rao for bringing the paper [START_REF] Debbah | Multiplicative free Convolution and Information-Plus-Noise Type Matrices[END_REF] to his attention and Øyvind Ryan and Serban Belinschi for some useful discussions. He would also like to thank an anonymous referee for suggesting him Remark 8.

1.

A relation between the Marchenko-Pastur law, the square and the rectangular free convolutions 1.1. Prerequisites on square and rectangular analytic transforms of probability measures.

1.1.1. The square case: the Rand S-transforms. These are analytic transforms of probability measures which allow to compute the operations ⊞ and ⊠, like the Fourier transform for the classical convolution. The R-transform can be defined for any probability measure on the real line, but we shall only define it for probability measures on [0, +∞). Consider such a probability measure µ.

It µ = δ 0 , then R µ = S µ = 0. Now, let us suppose that µ = δ 0 . Let us define the function M µ (z) = t∈R tz 1 -tz dµ(t).
Then the Rand S-transforms6 of µ, denoted respectively by R µ and S µ are the analytic functions defined as follows

(2) R µ (z) = [(1 + z)M -1 µ (z)] -1 , S µ (z) = 1 + z z M -1 µ (z),
where the exponent -1 refers to the inversion of functions with respect to the operation of composition •. Note that M µ is an analytic function defined in {z ∈ C ; 1/z / ∈ support(µ)}. Hence in the case where µ is compactly supported, the functions M µ and (1+z)M -1 µ (z) can be inverted in a neighborhood of zero as analytic functions in a neighborhood of zero vanishing at zero, with non null derivative at zero. In the case where µ is not compactly supported, these functions are inverted as functions on intervals (-ǫ, 0) which are equivalent to (positive constant) × z at zero [START_REF] Bercovici | Free convolution of measures with unbounded supports Indiana[END_REF].

Note that puting together both equations of (2), one gets

(3) S µ (z) = 1 z R -1 µ (z) = 1 + z z M -1 µ (z).
The main properties of the Rand S-transforms are the fact that they characterize measures and their weak convergence and that they allow to compute free convolutions : for all µ, ν,

(4) R µ⊞ν = R µ + R ν and S µ⊠ν = S µ S ν .
1.1.2. The rectangular case: the rectangular R-transform with ratio λ. In the same way, for λ ∈ [0, 1], the rectangular free convolution with ratio λ can be computed with an analytic transform of probability measures. Let ν be a symmetric probability measure on the real line.

Let us define H ν (z) = z(λM ν 2 (z) + 1)(M ν 2 (z) + 1), where ν 2 denotes the push forward of ν by the map t → t 2 . Then with the same conventions about inverses of functions than in the previous section, the rectangular R-transform with ratio λ of ν is defined to be

C ν (z) = U z H -1 ν (z) -1 , where U (z) = -λ-1+[(λ+1) 2 +4λz] 1/2 2λ
for λ > 0 and U (z) = z for λ = 0. By Theorems 3.8, 3.11 and 3.12 of [B09], the rectangular R-transform characterizes measures and their weak convergence, and for all pair ν 1 , ν 2 of symmetric probability measures, ν 1 ⊞ λ ν 2 is characterized by the fact that (5)

C ν 1 ⊞ λ ν 2 = C ν 1 + C ν 2 .
1.2. A relation between the square and the rectangular R-transforms. Let us fix λ ∈ [0, 1]. We recall that for any probability measure ρ on [0, +∞), √ ρ denotes the symmetrization of the push-forward of ρ by the map t → √ t and that for λ > 0, we have defined µ λ to be the law of λ times a random variable with law the Marchenko-Pastur law with mean 1/λ, i.e. µ λ is the law with support

[(1 - √ λ) 2 , (1 + √ λ) 2 ] and density x → 4λ -(x -1 -λ) 2 2πλx .
For λ = 0, we let µ λ denote the Dirac mass at 1.

Theorem 1. Let µ be a probability measure on [0, +∞). Then we have

R µ (z) = C √ µ⊠µ λ (z).
Remark 2 (The cumulants point of view). Suppose µ to be compactly supported. Let us denote the free cumulants [START_REF] Nica | Lectures on the combinatorics of free probability[END_REF] of µ by (k n (µ)) n≥1 and the rectangular free cumulants with ratio λ

[B09, B07b] of √ µ ⊠ µ λ by (c 2n ( √ µ ⊠ µ λ )) n≥1 .
Then the previous theorem means that for all n ≥ 1, one has

(6) k n (µ) = c 2n ( µ ⊠ µ λ ).
Proof.

-First of all, note that by continuity of the applications µ → µ⊠µ λ , ρ → R ρ and ν → C ν with respect to weak convergence [START_REF] Bercovici | Free convolution of measures with unbounded supports Indiana[END_REF][START_REF] Benaych-Georges | Rectangular random matrices, related convolution[END_REF], it suffices to prove the result in the case where µ is compactly supported. In this case, the functions

M µ , R µ , S µ , M µ⊠µ λ , H √ µ⊠µ λ , C √ µ⊠µ λ
are analytic in a neighborhood of zero and the operations of inversion on these functions or related ones can be used without precaution.

-If λ > 0, the free cumulants of the Marchenko-Pastur law with mean 1/λ are all equal to 1/λ, thus the ones of µ λ are given by the formula k n (µ λ ) = λ n-1 for all n ≥ 1 and R µ λ (z) = n≥1 λ n-1 z n . From (3), it follows that S µ λ (z) = 1 1+λz . Hence by (4), we have

S µ⊠µ λ (z) = Sµ(z)
1+λz , and by (3), ( 7)

M µ⊠µ λ (z) = M -1 µ (z) 1 + λz -1 .
Note that since µ 0 = δ 1 , (7) stays true if λ = 0. Now, let us define the function T (x) = (λx + 1)(x + 1). Note that T (U (x -1)) = x for x in a neighborhood of zero. We have

H √ µ⊠µ λ (z) = z × T • M µ⊠µ λ (z) = z × T • M -1 µ (z) 1 + λz -1 , and (8) C √ µ⊠µ λ (z) = U      z z × T • M -1 µ (z) 1+λz -1 -1 -1     
.

-Hence by (2) and (8), we have the following equivalence

R µ = C √ µ⊠µ λ ⇐⇒ (z + 1)M -1 µ (z) -1 = U      z z × T • M -1 µ (z) 1+λz -1 -1 -1      ⇐⇒ T • (z + 1)M -1 µ (z) -1 = z z × T • M -1 µ (z) 1+λz -1 -1 ⇐⇒   z × T • M -1 µ (z) 1 + λz -1   -1 × T • (z + 1)M -1 µ (z) -1 = z.
Composing both terms on the right by (z + 1)M -1 µ (z), it gives

R µ = C √ µ⊠µ λ ⇐⇒   z × T • M -1 µ (z) 1 + λz -1   -1 • ((z + 1)M -1 µ (z)) × T (z) = (z + 1)M -1 µ (z).
Dividing by T (z), it gives

R µ = C √ µ⊠µ λ ⇐⇒   z × T • M -1 µ (z) 1 + λz -1   -1 • ((z + 1)M -1 µ (z)) = M -1 µ (z) 1 + λz ⇐⇒ (z + 1)M -1 µ (z) =   z × T • M -1 µ (z) 1 + λz -1   • M -1 µ (z) 1 + λz ⇐⇒ (z + 1)M -1 µ (z) = M -1 µ (z) 1 + λz T (z),
which is obviously true by definition of T (z).

1.3. Main result of the paper. The main theorem of this paper is the following one. λ ∈ [0, 1] is still fixed.

Theorem 3. For any pair µ, µ ′ of probability measures on [0, +∞), we have

(9) µ ⊠ µ λ ⊞ λ µ ′ ⊠ µ λ = (µ ⊞ µ ′ ) ⊠ µ λ .
Remark 4. Note that in the case where λ = 0, this theorem expresses what we already knew about ⊞ 0 (and which is explained in the second paragraph of the introduction), but that the case λ = 1 isn't a consequence of the already known formula ⊞ 1 = ⊞.

Remark 5. Part of this theorem could have been deduced from Theorem 6 of [START_REF] Debbah | Multiplicative free Convolution and Information-Plus-Noise Type Matrices[END_REF]. However, (9) could be deduced from the theorem of Debbah and Ryan only for laws µ, µ ′ which can be expressed as limit singular laws of n by p (for n/p ≃ λ) corners of large p×p bi-unitarily invariant random matrices, but it follows from Theorem 14.10 of [START_REF] Nica | Lectures on the combinatorics of free probability[END_REF] that not every law has this form. Moreover, even though the idea which led us to our result was picked in the pioneer work of Debbah and Ryan, our proof is much shorter and shows the connection with the rectangular machinery in a more clear way (via Theorem 1 and Remark 2).

Proof. Define ν := √ µ ⊠ µ λ ⊞ λ √ µ ′ ⊠ µ λ . By (5), we have

C ν = C √ µ⊠µ λ + C √ µ ′ ⊠µ λ .
Thus, by Theorem 1, and (4), we have

C ν = R µ + R µ ′ = R µ⊞µ ′ = C √ (µ⊞µ ′ )⊠µ λ .
Hence by injectivity of the rectangular R-transform (Theorem 3.8 of [B09]), ( 9) is valid.

The formula (9) gives us a new insight on rectangular free convolutions: it allows to express it, in certain cases, in terms of the free convolutions "of square type" ⊞ and ⊠. However, only laws which can be expressed under the form (10) µ ⊠ µ λ , (µ probability measure on [0, +∞))

can have their rectangular convolution computed via formula (9). Thus it seems natural to ask whether all symmetric laws can be expressed like in (10). Note that it is equivalent to the fact that any law on [0, +∞) can be expressed under the form µ ⊠ µ λ , which is equivalent to the fact that the Dirac mass at one δ 1 can be expressed under the form µ ⊠ µ λ . Indeed, if δ 1 = µ ⊠ µ λ , then any law τ on [0, +∞) satisfies τ = τ ⊠ δ 1 = (τ ⊠ µ) ⊠ µ λ . The following proposition shows that it is not the case. However, Theorem 11 will show that many symmetric laws can be expressed like in (10).

and µ 1 = µ. For all t, µ t is denoted by µ * t (resp. µ ⊞t , µ ⊞ λ t ). Infinitely divisible distributions have been classified: µ is * -(resp. ⊞-) infinitely divisible if and only if there exists a real number γ and a positive finite measure on the real line σ such that the Fourier transform is μ

(t) = exp iγt + R (e itx -1 -itx x 2 +1 ) x 2 +1 x 2 dσ(x) (resp. R µ (z) = γz +z R z+t 1-tz dσ(t))
. Moreover, in this case, such a pair (γ, σ) is unique, it is called the Lévy pair of µ and we denote µ by ν γ,σ * (resp. ν γ,σ ⊞ ). For all t ≥ 0, µ t has Lévy pair (tγ, tσ). In the same way, a symmetric probability measure ν is ⊞ λ -infinitely divisible if and only if there exists a positive finite symmetric measure on the real line

G such that C ν (z) = z R 1+t 2
1-zt 2 dG(t). In this case, the measure G is unique, and ν will be denoted by ν G ⊞ λ . The correspondences ν γ,σ * ←→ ν γ,σ ⊞ (for any pair (γ, σ) as above) and ν 0,G * ←→ ν G ⊞ λ (for any G as above) are called Bercovici-Pata bijections. These bijections have many deep properties [START_REF] Bercovici | Stable laws and domains of attraction in free probability theory[END_REF][START_REF] Benaych-Georges | Infinitely divisible distributions for rectangular free convolution: classification and matricial interpretation Probability Theory and Related Fields[END_REF], some of which will be mentioned in the proof of the following lemma.

Lemma 9. Let γ be a real number and σ be a positive finite measure on the real line. Then we have equivalence between:

(i) For all t ≥ 0, ν tγ,tσ * is supported on [0, +∞).

(ii) For all t ≥ 0, ν tγ,tσ ⊞ is supported on [0, +∞).

(iii) We have σ((-∞, 0]) = 0 and the integral 1

x dσ(x) is finite and ≤ γ.

Proof. The equivalence between (i) and (iii) follows from Theorem 24.7 and Corollary 24.8 of [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF]. Let us prove the equivalence between (i) and (ii). In order to do that, let us recall a fact proved in [START_REF] Bercovici | Stable laws and domains of attraction in free probability theory[END_REF]: for any Lévy pair (γ, σ) and any sequence (ν n ) of probability measures, one has (13) ν * n n converges weakly to ν γ,σ * ⇐⇒ ν ⊞n n converges weakly to ν γ,σ ⊞ . Let us suppose (i) (resp. (ii)) to be true. Let us fix t ≥ 0. For all n, we have Thus since any free (resp. classical) additive convolution and any weak limit of measures with supports on [0, +∞) has support on [0, +∞), (ii) (resp. (i)) holds.

Remark 10. Note that (i) is equivalent to the fact that there exists t > 0 such that ν tγ,tσ * is supported on [0, +∞) [Sa99, Cor. 24.8]. However, the same is not true for the free infinitely divisible laws. Indeed, let, for each t ≥ 0, MP t denote the Marchenko-Pastur law with mean t [HP00, Ex. 3.3.5] and let us define, for each t, µ t = MP t * δ -t/4 . Then since free and classical convolutions with Dirac masses are the same, (µ t ) t≥0 is a convolution semi-group with respect to ⊞. But µ 4 is supported on [0, +∞), whereas for each t ∈ (0, 1], the support of µ t contains a negative number (namely -t/4).

2.2.

Main result of the section. The following theorem allows us to claim that even though not every symmetric law can be expressed under the form √ µ ⊠ µ λ for µ law on [0, +∞) (see Proposition 6), many of them have this form. λ ∈ [0, 1] is still fixed.

For G measure on the real line, we let G 2 denote the push-forward of G by the function t → t 2 . Theorem 11. (i) Let µ be a ⊞-infinitely divisible law such that for all t ≥ 0, µ ⊞t is supported on [0, +∞). Then the law √ µ ⊠ µ λ is ⊞ λ -infinitely divisible, with Lévy measure the only symmetric measure G such that

(14) G 2 = γ - 1 x dσ(x) δ 0 + 1 + x 2 x(1 + x) dσ(x),
where (γ, σ) denotes the Lévy pair of µ.

(ii) Reciprocally, any ⊞ λ -infinitely divisible law ν has the form √ µ ⊠ µ λ for some ⊞-infinitely divisible law µ such that for all t ≥ 0, µ ⊞t is supported on [0, +∞). Moreover, the Lévy pair (γ, σ) of µ is defined by

(15) γ = [0,+∞) 1 + x 1 + x 2 dG 2 (x) and σ = x(1 + x) 1 + x 2 dG 2 (x),
where G denotes the Lévy measure of ν.

Proof. (i) Note that by Theorem 3, the map µ → √ µ ⊠ µ λ is a morphism from the set of laws on [0, +∞) to the set on symmetric laws on the real line endowed respectively with the operations ⊞ and

⊞ λ . Thus if µ is ⊞-infinitely divisible, then √ µ ⊠ µ λ is ⊞ λ -infinitely divisible. Moreover, if the Lévy pair of µ is (γ, σ), then its R-transform is R µ (z) = γz + z t∈R z+t 1-zt dσ(t). By Theorem 1, it implies that C √ µ⊠µ λ (z) = γz + z t∈R z+t 1-zt dσ(t). But by uniqueness, the Lévy measure G of √ µ ⊠ µ λ is characterized by the fact that C √ µ⊠µ λ (z) = z R 1+t 2
1-zt 2 dG(t). Thus to prove (14), it suffices to prove that for G given by ( 14), for all z, one has

γz + z t∈R z + t 1 -zt dσ(t) = z R 1 + t 2 1 -zt 2 dG(t),
which can easily be verified.

(ii) Let ν be a ⊞ λ -infinitely divisible law with Lévy measure denoted by G. Let (γ, σ) be the Lévy pair defined by (15). Note that (γ, σ) satisfies (iii) of Lemma 9, thus, for µ := ν γ,σ ⊞ , for all t ≥ 0, the law µ ⊞t is actually supported by [0, +∞). Thus by (i), √ µ ⊠ µ λ is ⊞ λ -infinitely divisible with Lévy measure the only symmetric measure H satisfying

H 2 = γ - 1 x dσ(x) δ 0 + 1 + x 2 x(1 + x) dσ(x).
To prove that √ µ ⊠ µ λ = ν, it suffices to prove that H = G, which can easily be verified.

One of the consequences of this theorem is that it gives us a description of the free multiplicative convolution of two Marchenko-Pastur laws (i.e. free Poisson laws), one of them having a mean ≥ 1. For all t > 0, the Marchenko-Pastur law MP t with mean t has been introduced at Remark 10.

Corollary 12. Consider a, c > 0 such that a > 1. Then MP c ⊠ MP a is the push forward, by the map x → ax 2 , of the ⊞ λ -infinitely divisible law with Lévy measure c 4 (δ 1 + δ -1 ) for λ = 1/a.

Proof. It suffices to notice that for λ = 1/a, MP a is the push-forward, by the map x → ax, of the law µ λ , that MP c is the ⊞-infinitely divisible law with Lévy pair (c/2, c/2δ 1 ), and then to apply (i) of Theorem 11.

This corollary can be interpreted as the coincidence of the limit laws of two different matrix models. Indeed, the ⊞ λ -infinitely divisible law with Lévy measure c 4 (δ 1 + δ -1 ) was already known [B07a, Prop. 6.1] to be the limit symmetrized singular law of the random matrix M := p k=1 u k v * k , for n, p, q tending to infinity in such a way that p/n → c and n/q → λ and (u k ) k≥1 , (v k ) k≥1 two independent families of independent random vectors such that for all k, u k , v k are uniformly distributed on the unit spheres of respectively C n , C q . Thus, if, for large n, p, q's such that p/n ≃ c and n/q ≃ λ, one considers such a random matrix M and also two independent random matrices T, Q with respective dimensions n×p, n×q, the entries of which are independent real standard Gaussian random variables, then the empirical spectral measures of the random matrices M M * and 1 nq T T * QQ * are close to each other, as illustrated by Figure 1.

Introduction 1 .

 1 A relation between the Marchenko-Pastur law, the square and the rectangular free convolutions 1.1. Prerequisites on square and rectangular analytic transforms of probability measures 1.2. A relation between the square and the rectangular R-transforms 1.3. Main result of the paper 2. Consequences on square and rectangular infinite divisibility 2.1. Prerequisites on infinite divisibility and Lévy-Kinchine formulas 2.2. Main result of the section References Introduction

)

  ⊞n converges weakly to ν tγ,tσ ⊞ (resp. (ν tγ n , tσ n ⊞ ) * n converges weakly to ν tγ,tσ *).

Figure 1 .

 1 Figure 1. Histograms of the spectrums of M M * (left) and 1 nq T T * QQ * (right) for n = 2000, λ = 0.6, c = 1.3.

See the footnote 5.

This work was partially supported by the Agence Nationale de la Recherche grant ANR-08-BLAN-0311-03.

2

Proposition 6. Unless λ = 0, the law δ 1 +δ -1 2 cannot be expressed under the form √ µ ⊠ µ λ for µ probability measure on [0, +∞).

Proof.

Suppose that λ > 0 and that there is a probability measure µ on [0, +∞) such that δ 1 +δ -1 2 = √ µ ⊠ µ λ . Then δ 1 = µ ⊠ µ λ . This is impossible, by Corollary 3.4 of [B06],

which states that the free multiplicative convolution of two laws which are not Dirac masses has always a non null absolutely continuous part (there is another, more direct way to see that it is impossible: by (4), such a law µ has to satisfy S µ (z) = 1 + λz, which implies that for z small enough, M µ (z) = z-1+[(1-z) 2 +4λz] 1/2 2λ : such a function doesn't admit any analytic continuation to C\[0, +∞), thus no such probability measure µ exists).

Theorem 3 has a consequence on the free convolutions "of square type" which wasn't known yet, despite the many papers written the last years about questions related to the arithmetics of these convolutions, e.g. [BV95, BPB99, B04, CG08a, CG08b, BBG08, BBCC08].

Corollary 7. For any pair µ, µ ′ , of probability measures on [0, +∞) we have

Proof. It is an obvious consequence of Theorem 3 and of the fact that ⊞ 1 = ⊞.

Remark 8. The referee of the paper communicated to us a proof of (11) which is not, as ours, based on computations on the Rand S-transforms, but on the direct proof of (6) in the special case λ = 1. Let us briefly outline this proof. When λ = 1, by [B07a, Eq. (4.1)], (6) reduces to

Let a, s are free elements in a tracial non commutative probability space with respective distributions µ and the standard semicircle law. By [NS06, Prop. 12.13], s 2 has distribution µ 1 , hence sas has distribution µ ⊠ µ 1 . It follows, by [NS06, Prop. 12.18], that for all n, the n-th moment of µ is equal to k n (µ ⊠ µ 1 ). But by [START_REF] Nica | Lectures on the combinatorics of free probability[END_REF]Prop. 11.25], for all n, we have

It follows, using the expression of the n-th moment of µ in terms of its free cumulants, that for all n,

and that for all n, k n (µ) = k 2n ( √ µ ⊠ µ 1 ).

Consequences on square and rectangular infinite divisibility

2.1. Prerequisites on infinite divisibility and Lévy-Kinchine formulas. Infinite divisibility is a fundamental probabilistic notion, at the base of Lévy processes, and which allows to explain deep relations between limit theorems for sums of either independent random variables, square or rectangular random matrices. Let us briefly recall basics of this theory [GK54, Sa99, BV93, BPB99, B07a].

Let * denote the classical convolution of probability measures on the real line. Firstly, recall that a probability measure µ is said to be * -infinitely divisible (resp. ⊞-, ⊞ λ -infinitely divisible) if for all integer n, there exists a probability measure ν n such that ν * n n = µ (resp. ν ⊞n n = µ, ν ⊞ λ n n = µ). In this case, there exists a * -(resp. ⊞-, ⊞ λ -) semigroup (µ t ) t≥0 such that µ 0 = δ 0